Conceptual Neighbourhood Diagrams for

Conceptual Neighbourhood Diagrams for Representing Moving Objects Nico Van de Weghe and Philippe De Maeyer Department of Geography, Ghent University, Krijgslaan 281 (S8), B-9000 Ghent, Belgium Abstract. The idea of Conceptual Neighbourhood Diagram (CND) has proved its relevance in the areas of qualitative reasoning about time and qualitative reasoning about space. In this work, a CND is constructed for the Qualitative Trajectory Calculus (QTC), being a calculus for representing and reasoning about movements of objects. The CND for QTC is based on two central concepts having their importance in the qualitative approach: the theory of dominance and the conceptual distance between qualitative relations. Some examples are given for illustrating the use and the potentials of the CND for QTC from the point of view of GIScience. 1. Introduction Since humans usually prefer to communicate in qualitative categories supporting their intuition and not in quantitative categories, qualitative relations are essential components of queries that people would like to run on a GIS. Temporal calculi, such as the Interval Calculus [1] and the Semi-Interval Calculus [2], have been proposed. In addition, spatial calculi, such as the RCC-calculus [3] and the 9-Intersection Model [4], both focusing on topological relations between regions, have been proposed. These topological relations form the basis of most current GISs. Despite extensive research during the past decade, both from the area of spatio-temporal reasoning (e.g. [5-7]) and spatio-temporal databases (e.g. [8-11]), the representation of space-time is problematic, and a full temporal GIS is not available yet. We believe that qualitative spatio-temporal calculi should take a central place behind such a temporal GIS. In [12], the Qualitative Trajectory Calculus is presented, being a theory for representing and reasoning about moving objects in a qualitative framework. Depending on the level of detail and the number of spatial dimensions, different types of this calculus are defined and studied in detail. In this paper, we focus on the basic Qualitative Trajectory Calculus in one dimension, QTC for short. In Section 2, QTC is presented. After a brief description concerning Conceptual Neighbourhood Diagrams (CND), this idea is applied to the domain of continuously moving objects in Section 3. We discuss two concepts, used for the construction of the CNDs for QTC: dominance space and conceptual distance. In Section 4, some examples on how to use the CND for QTC are presented. 2. The basic Qualitative Trajectory Calculus for One Dimension (QTC) QTC handles the qualitative movement of a pair of point objects along a 1D line, such as cars driving on the same lane and trains moving on a railroad. We assume continuous time for QTC.

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Nico Van de Weghe and Philippe De Maeyer

2.1 QTC of Level One (QTCL1) Because the movement is restricted to 1D, the velocity vector of an object is restricted to two directions, with the intermediate case where the object stands still. Hence, the direction of the movement of each object can be described by one qualitative variable, using the following conditions, resulting in 9 so-called L1-relations (Fig. 1).1 1. Movement of k, with respect to the position of l at time point t (distance constraint) −: k is moving towards l; +: k is moving away from l; 0: k is stable with respect to l. 2. Movement of l, with respect to the position of k at time point t (distance constraint) −: l is moving towards k; +: l is moving away from k; 0: l is stable with respect to k.

Fig. 1: L1-relation icons

2.2 QTC of Level Two (QTCL2) QTC L1 can be extended with a third character giving the relative speed of the objects: 3. Relative speed of k at time point t with respect to l at t (speed constraint) −: vk at t < vl at t; +: vk at t > vl at t; 0: vk at t = vl at t This way, we can create all combinations for QTCL2. 10 of the 27 theoretic possibilities are impossible, resulting in only 17 L2-relations (Fig. 2).2

Fig. 2: L2-relation icons

Note that QTC can handle as well movements of objects along a curved line. This is done by considering the distance and speed constraints along the curved line. As a direct result, situations where, for example, two cyclists are moving along a curved cycle track, can be handled in QTC.

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The left and right dot represent respectively the positions of k and l. The dashed line segments represent the potential object movements; the lines are dashed since there is no direct information about the speed of the objects. The line segments represent whether each object is moving towards or away from the other. A dot is filled if the object can be stationary. 2 Note that the lines can have different lengths giving the difference in relative speed.

Conceptual Neighbourhood Diagrams for Representing Moving Objects

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3. Conceptual Neighbourhood Diagram for QTC 3.1 Definition of a Conceptual Neighbourhood Diagram (CND) CNDs have been introduced in the temporal domain [2], and have been widely used in spatial reasoning, e.g.: for topological relations [3,13]; cardinal directions [14], and for relative orientation [15]. CNDs are typically used for qualitative simulation to predict what will happen in the future. Two relations between entities are conceptual neighbours, if they can be transformed into one another by continuously deforming, without passing another qualitative relation; a CND describes all the possible transitions between relations that can occur [2]. For clarification of these definitions, we use the CND for RCC (Fig. 3). The relations DC and EC are conceptual neighbours, since continuous deformation is possible by moving k and l towards each other. DC and PO are not conceptual neighbours, because a continuous deformation cannot transform from DC into PO without passing through EC.

Fig. 3: CND for topological relations in the RCC-calculus

3.2 Construction Concepts Based on [2], we define: two QTC relations are conceptual neighbours if they can directly follow each other during a continuous movement. We discuss two concepts for the construction of the CNDs for QTC: dominance space and conceptual distance. 3.2.1 Theory of Dominance Central in the theory of dominance [16,17] are the constraints imposed by continuity. Consider the qualitative distinction between −, 0 and +. A direct change from – to + is impossible, since such a change must pass the qualitative value 0, that only needs to hold for an instant. On the other hand, the + of a variable changing from 0 to + and back to 0, must hold over an interval. We say that 0 dominates − and + [18]. Now, let us focus on QTC. A change from the L1-relation (– +)L1 to (+ +)L1, must pass at least one QTC relation, since the first character cannot chance continuously from − to +. The shortest way is via (0 +)L1. This relation only needs to hold for an instant. On the other hand, the (+ +)L1 of the sequence of relations {(+ 0)x(+ +)x(+ 0)}L1, must hold over an interval.3 In order to explain this, consider (+ 0)L1 at t1, (+ +)L1 at t2, and the speed of l at t2 being 0.1 metres per second. One can always find a time point 3

The transition from a to b is denoted by axb.

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between t1 and t2 with the speed of l being somewhere in between 0 metres per second and 0.1 metres per second. In the words of Galton: "When an object starts moving, there is a last moment when it is at rest, but no first moment when it is in motion" [19, p.101]. Thus, (+ 0)L1 dominates (+ +)L1. Now, one can construct a dominance space, being a space containing qualitative values and their according dominance relations [16]. Fig. 4 represents a basic example of the dominance space in 1D: a transition from − to 0 can occur and vice versa (with 0 dominating −); a transition from 0 to + can occur and vice versa (with 0 dominating +); a transition from − to + can only occur by passing through 0. Fig. 4: Dominance space in 1D

It has been proved in [18] that simple dominance spaces can be combined for building composite dominance spaces. We use this theorem for the construction of the CNDs for QTC. Fig. 5a, for example, shows the composite dominance space in 2D, both dimensions containing the qualitative values 0 and +. Each dimension of this space, which can be seen as a subset of all L1-relations, contains two connections. Combining both dimensions, {(0 0)x(+ +)}L1 is constructed. This can be done by composing {(0 0)x(+ 0)}L1 with {(+ 0)x(+ +)}L1, or by composing {(0 0)x(0 +)}L1 with {(0 +)x(+ +)}L1. Note the importance of the direction of the connections. Based on [17] and Fig. 5b, we explain that it is impossible to construct {(0 +)x(+ 0)}L1. (0 +)L1 holds over an interval ]t1,t2[, and (+ 0)L1 holds over interval ]t2,t3[. Thus, the first character changes from 0 to + at t2, and the second character changes from + to 0 at t2. Since 0 dominates +, the relation at t2 will be (0 0)L1.

Fig. 5: Dominance space in 2D

3.2.2 QTC Distance (dQTC) Based on the ideas of topology distance [13] and distance measure between two cardinal directions [20], we define QTC distance (dQTC). The conceptual distance between qualitative relations is the shortest path between these relations in the CND, giving every arc a distance being equal to one. As a result, dQTC is the conceptual distance between two QTC relations. Let us consider three examples: - If R1 and R2 differ in one character that can change continuously between both states without passing through an intermediate qualitative value. E.g.: if R1 = (0 0)L1 and R2 = (0 +)L1 then dQTC = 1, denoted by: dQTC(0 0,0 +) = 1. - If R1 and R2 differ in one character that cannot change between both states without passing through an intermediate qualitative value, then dQTC is composed of sub-distances. E.g.: dQTC(0 −,0 +) = dQTC(0 −,0 0) + dQTC(0 0,0 +) = 1 + 1 = 2


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- If R1 and R2 differ in multiple characters, then dQTC is a composition of the sub-distances determined for each individual character. If multiple compositions are possible, then the composition resulting in the lowest dQTC is selected. E.g.: dQTC(− −,+ +) = dQTC for 1st character + dQTC for 2nd character = 2 + 2 = 4. 3.3 Conceptual Neighbourhood Diagram for QTCL2 (CNDL2)4 Based on the theory of dominance and QTC distance, we construct CNDL2. As proposed in [16], we start with simple dominance spaces, and build these into composite dominance spaces. First, consider the three dominance spaces with dQTC = 1 (one-dominance space) in Fig. 6a-c, representing three subsets of dominance relations: continuous changes of the first (a), second (b) and third (c) character.

Fig. 6: One-dominance spaces for QTCL2

Fig. 7: Construction of two-dominance spaces for QTCL2 4

Due to space limitations, we will not handle the CNDL1, studied in depth in [6].

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There are three possible compositions of one-dominance spaces, resulting in a twodominance space (Fig. 7a,b). There exists a second composition level in which each two-dominance space is combined with a one-dominance space (being orthogonal to both dimensions of the two-dominance space). Fig. 8a shows the three different composition possibilities resulting in the same three-dominance space (Fig. 8b).

Fig. 8: Construction of three-dominance spaces for QTCL2

The disjunction of the 3 one-dominance spaces, the 3 two-dominance spaces and the three-dominance space, results in the overall dominance space (Fig. 9).

Fig. 9: Overall-dominance space for QTCL2

A clearer view of the dominance space can be obtained by deleting all the 'impossible' nodes and rearranging those that remain [16]. By deleting the 10


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impossible labels, and the according arcs starting from or ending in one of these 10 relations, CNDL2 is created, represented in 3D (Fig. 10a) and in 2D (Fig. 10b).

Fig. 10: CNDL2

CNDL2 forms the basis for two CND icons: the static CNDL2 icon (Fig. 11) represents a set of L2-relations, the dynamic CNDL2 icon (Fig. 13) represents a sequence of relations, and will be explained in detail in the next section. {−−−}L2

−−0}L2

{

{−−+, −+ 0}L2

Fig. 11: Static CNDL2 icon

4. Examples of CND for QTC 4.1 Multiple Time Points

Fig. 12: History of multiple point objects in 1D

Consider k and l moving in 1D, during a study period between t1 and t9 (Fig. 12). Nothing is known about the situations immediately before t1 and immediately after t9. Therefore, it is impossible to label t1 and t9. The L2-relations for i1, t2, i2, …, i7, t8, and i8 can be created quite easily by use of some specific rules: - 0 dominates + and –. E.g.: t7 is between i6(– – 0)L2 and i7(– – +)L2. Thus, t7(– – 0)L2.

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- Transition from – to + (and vice versa) is impossible without passing 0.5 E.g.: Distance constraint: transition from i7(– – +)L2 to i8(– + +)L2 is impossible without passing through t8(– 0 +)L2. Speed constraint: transition from i2(+ + –)L2 to i3(+ + +)L2 is impossible without passing through t3(+ + 0)L2. Relations t8(– 0 –)L2 and t3(+ + 0)L2 only hold for an instantaneous time point. One should be aware that the qualitative value 0 can also hold for an interval, e.g. i6(– – 0)L2. Be also aware that it is possible that multiple characters change simultaneously. This is for example the case for {t6(0 0 0)xi6(– – 0)}L2. - Combination of both former rules. E.g.: t2 is between i1(– 0 +)L2 and i2(+ + –)L2. Thus, t2(0 0 0)L2. Finally, CNDL2 can be drawn (Fig. 13), with the labels and the directed arcs respectively referring to the temporal primitives and the transitions. If there is an absence of transition between multiple temporal primitives, all these units are written as has been done for '(0 0 0): t5, i5, t6'. Note that the same node can be passed multiple times during a study period.

i2 t3 i8

i3 i1 t8

t2 t 5,i 5,t 6

t4

i7

i4 i 6,t 7

Fig. 13: Dynamic CNDL2 icon

4.2 Incomplete Knowledge about Moving Objects This example starts from an expression (Ex1) forming fine knowledge concerning moving objects, and relaxes the constraints in order to get incomplete knowledge (Ex2 and Ex3). Thereafter, we work the other way round. 4.2.1 From Fine to Incomplete Knowledge - Ex1: k is moving towards l, which in turn is moving away from k, both objects having the same speed. (Fig. 14a) Ex1 ↔ (– + 0)L2 - Ex2: k is moving towards l, which in turn is moving away from k. (Fig. 14b) Ex2 ↔ (– + –, – + 0, – + +)L2 - Ex3: k is moving towards l. (Fig. 14c) Ex3 ↔ (– – –, – – 0, – – +, – 0 –, – 0 0, – 0 +, – + –, – + 0, – + +)L2 5

This counts for the direction as well as the speed constraint.


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(a) (b) (c) Fig. 14: Static CNDL2 icon for (a) Ex1, (b) Ex2 and (c) Ex3

4.2.2 From Incomplete to Fine Knowledge Now, let us start from 3 expressions (Ex1a, Ex1b and Ex1c), which together form the fine compound expression Ex1: - Ex1a: k is moving towards l. (Fig. 15a) Ex1a ↔ (– – –, – – 0, – – +, – 0 –, – 0 0, – 0 +, – + –, – + 0, – + +)L2 - Ex1b: l is moving away from k. (Fig. 15b) Ex1b ↔ (– + –, – + 0, – + +, 0 + –, 0 + 0, 0 + +, + + –, + + 0, + + +)L2 - Ex1c: k and l have the same speed. (Fig. 15c) Ex1c ↔ (– – 0, – + 0, 0 0 0, + – 0, + + 0)L2 The conjunction of the solutions of Ex1a, Ex1b and Ex1c gives (– + 0)L2 (Fig. 15d).

(a) (b) (c) (d) Fig. 15: Static CNDL2 icon for (a) Ex1a, (b) Ex1a, (c) Ex1c, and (d) Ex1

By comparing Fig. 15d with Fig. 14a, one can state that the conjunction of the solution sets of the components of a compound expression is the same as the solution set of the compound expression. 5. Summary and Further Work In this work, a conceptual neighbourhood diagram (CND) was constructed for the Qualitative Trajectory Calculus (QTC), being a calculus for representing and reasoning about movements of objects. After a brief description concerning CNDs, this idea was applied to the domain of continuously moving objects. We discussed two concepts, used for the construction of the CNDs for QTC: dominance space and conceptual distance. The CND for QTC forms the basis for the static CND icon (representing QTC relations) and the dynamic CND icon (representing a sequence of QTC relations). Finally, some examples on how to use the CND for QTC were presented. We believe that, apart from a neat visualisation, the CND and its icons can represent specific types of conceptual behaviour, which could lead to conceptual modelling and qualitative simulation of moving objects. The CND for QTC is specifically well-suited for reasoning about incomplete knowledge of moving objects. This work is part of a larger research question that can be formulated as: 'how to describe motion adequately within a qualitative calculus, so as to obtain a tool for data and knowledge representation and for querying spatio-temporal data'. A full answer to this question needs, besides the spatio-temporal reasoning, also an exhaustive study of several database issues, increasing general performance by the use of efficient algorithms and access methods for computing intensive query operations.

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