In a related work, we propose a model for the description of relationships in space and time, that is, an algebra which identifies spatio-temporal relationships between independent entities (Claramunt and Jiang 2000). In fact, the modelling of relationships that integrate both the temporal and spatial dimensions is still a challenge for GIS research. A closer investigation of the temporal dimension has shown that hierarchical analysis of events and states in time is often employed in many GIS application areas (Whigham 1993). Such a hierarchical representation is based on the fact that events are often referred and organised using a calendar (e.g., year-month-day) that reflects the cyclic structure of time. A calendar introduces a different range of relationships than those usually identified within linear-time algebra: for example two daily events, disjoint on a linear time-line can be closely related within a cyclic representation of time (e.g., July 14, 1998 and July 14, 1999). Therefore, we believe that the analysis of temporal relationships between events are not limited to a strict application of basic operations between temporal intervals in a linear time-line, but instead to an analysis of the "proximity" of these temporal intervals at different levels of the underlying calendar hierarchy that represents the cycle of time. Similarly, the study of spatial relationships can be developed using a nested hierarchical but not cyclic organisation of regions in space (Kainz et al. 1993, Car and Frank 1994, Kuipers 1996). For example, two disjoint buildings may or may not belong to the same neighbourhood at a higher level of abstraction. Our model proposes a representation of temporal and topological relationships using a hierarchical and cyclic model of the time-line and a hierarchical model of the twodimensional space. We show that such a hierarchical approach provides a flexible set of temporal and topological relationships that together provide some nice properties for many GIS applications. The concepts presented in this research are illustrated in the context of a fictive crime study. The remainder of this paper is organised as follows. Section 2 introduces the principles of our model. Section 3 develops the formal representation of hierarchical temporal relationships. Section 4 applies similar hierarchical concepts to spatial reasoning. Finally section 5 concludes the paper.

2. MODEL PRINCIPLES Current temporal reasoning algebras are based, to the best of our knowledge, on a linear representation of time (Allen 1984). Time is represented as a set of measured times isomorphic to a set of real numbers. As in Allen's algebra, we consider convex periods as the basic temporal primitive of our model. We define a period as an anchored duration of time delimited by two time-stamps, an interval as the length of a period (Snodgrass et al. 1995). Allen's algebra defines 7 temporal relationships (13 with their inverses) between temporal periods (Figure 1).

i1 before i2 i1 meets i2 i1 o verlaps i2 i1 equal i2

i1 during i2 i1 starts i2 i1 finishes i2

time line

time period i1 time period i2

Figure 1: Allen's temporal relationships

2

This algebra is widely used in temporal reasoning. It has also been considered as a reference, or at least as an equivalent, by temporal database models and query languages (Snodgrass et al. 1995). However, we believe that a linear view of the time-line is not sufficient for an analysis of event patterns in space and time. Many application domains widely use a cyclic and hierarchical representation of the temporal dimension. Such applications are based on the concept of calendar, that is, a hierarchical subdivision of time (Figure 2). A calendar is defined by a hierarchy of non-intersecting temporal intervals. Each temporal interval of the hierarchy can be decomposed by a sum of temporal intervals defined at the immediate lower level of the hierarchy, if any (i.e., does not apply for the lowest level of the hierarchy). We denote a calendar as follows: · Let C(A1, A2, ..., Am) be a calendar where A1, A2, ..., Am represent the temporal intervals of the calendar hierarchy and m the number of levels of the calendar hierarchy, also called the depth of the calendar (Chandra et al. 1998). In general, there are two alternatives to the above notation: either A1 represents the finest temporal interval level of the hierarchy and Am the coarsest temporal interval level of the hierarchy, or in contrast, A1 represents the coarsest temporal interval level of the hierarchy and Am the finest temporal interval level of the hierarchy. Most Western calendars adopt the former approach, whilst most Eastern calendars take the latter approach. We will take the second option in the remainder of this paper. Let us consider the example of a calendar defined with years, months and days. A year is decomposed in 12 months, a month in either 28, 29, 30 or 31 days (weeks lead generally to some problems in the representation of calendars as the same week can belong to two different months or even years in some cases). The notion of granularity is closely related to the concept of calendar. Granularity is one of the key concepts in temporal database modelling. The granularity of an event can be defined as the smaller temporal interval unit used in the modelling of this event. For instance, the concepts of being early, being late and being at the same time, are strongly affected by the level of granularity used. The required precision of the time-axis depends upon the application. For very dynamic phenomena, minute and second-based granularity units are often used, whilst dealing with more conventional applications, a calendar time scale based upon monthly, weekly or daily granularity units is more relevant. Year

January Feburary 1, 2, ... 31 1, 2, ... 28

... ...

December 1, 2, ... 31

1:00, 2:00, ... 24:00

...

1:00, 2:00, ... 24:00

Figure 2: Example of temporal hierarchy

Although some events may be instantaneous, most occur over a period of time (Allen 1991). Accordingly, we give a temporal period extension to an event as this corresponds more closely to real world scenario. An event can be defined as an application-driven concept that supports a cognitive interpretation of a significant pattern of change. The concepts of states and events are often related as discussed in Allen (1991). Our model is

3

oriented to the manipulation of events, it also applies to states as far as these are represented using convex temporal intervals and a calendar. We define an event as follows: · Let e1(a1, a2, ..., ap) be an event defined using a calendar C(A1, A2, ..., Ap, ..., Am) where a1, a2, ..., ap represent temporal periods (limited to convex periods within the scope of our model) defined using the temporal intervals of the calendar hierarchy, that is, A1, A2, ..., Ap, respectively ; p is the number of temporal period levels of e1, 1 p m , ap ap-1 ... a1. Ap gives the granularity of e1. Our model does not attempt to provide a general theory for hierarchical reasoning in time. We limit the scope of our framework to short events whose each temporal period is "contained" (i.e., either during, starts, finishes or equal) in a temporal period defined by an interval unit of the immediate superior level of the calendar hierarchy (i.e., nested hierarchy). Let us illustrate this concept with a calendar C1(Month, Day, Hour). For example, an event defined by a granularity at the finest level of this calendar (i.e., Hour) is "contained" in a period unit defined at the immediate superior level of the calendar hierarchy, that is, a day period. Therefore, temporal intervals represented at the finest level of such a calendar are restricted to a 24 hour period of reference (e.g., [10:00, 18:00]). Such a restriction applies very well to the study of daily events generally used for the analysis of crime patterns, monitoring of urban traffic conditions or some environmental studies, it is also required to reflect the properties of the nested hierarchy. Hierarchical reasoning is not restricted to the temporal dimension, space can also be formalised and analysed using a hierarchical view (Hirtle and Jonides 1985, Whigham 1993, Kainz et al. 1993, Car 1994, Kuipers 1996). We believe that such a principle is adapted to many domains in which possible relationships between located events involve both temporal and topological relationships. We consider daily events located in space. We model space as a nested hierarchy of non-intersecting regions. Each region of the hierarchy can be recursively decomposed (i.e., dis-aggregated) by several regions of space at the immediate lower level if any of that hierarchy. Using such a hierarchy, each event in space can be recursively decomposed into several regions of space. Administrative and political subdivisions are often organised using such a hierarchy. Let us take the example of an urban environment, that is, a city composed of a set of non-intersecting districts, each district composed of a set of non-intersecting neighbourhoods, and each neighbourhood composed of a set of non-intersecting buildings. Such a spatial hierarchy can be illustrated graphically as in Figure 3. City

District(1)

District(2)

...

District(n)

Neighb.(1) Neighb.(2) ...

...

...

...

...

Neighb .(p)

Build . ( 1 ) Build. ( 2 ) ... ... ... Figure 3: Example of spatial hierarchy

......

Build.(q)

We define a spatial hierarchy as follows: · Let SH( S1 , S2 , . . . , Sn ) be a spatial hierarchy where S1 , S2 , . . . , Sn represent successive levels of abstraction in the spatial dimension, from the highest, that is S1, to

4

the lowest, that is Sn, levels of abstraction, and n the number of levels of abstraction of SH. Then let us define an event located in space: · Let e1(a1, a2, ..., ap) be an event defined using a calendar C(A1, A2, ..., Ap, ..., Am). We say that e1 is hierarchically located in space iff there is a spatial hierarchy SH( S1 , S2 , ... , S q , ... , Sn ) such as es1 (s1, s2, ..., sq) represents the nested regions of SH that recursively locate e1 in space, 1 q n , s1 (equal covers contains) s2, ..., sq-1 (equal covers contains) sq. The latter topological constraints represent an intrinsiC Property of the spatial hierarchy (i.e., non-intersecting regions). They are defined using well-known topological relationships in two-dimensional spaces (Egenhofer 1991). This nested spatial hierarchy includes non-intersecting regions at each of its levels. Therefore, topological relationships between regions that respectively represent some events at a same level of the hierarchy are either disjoint, touch or equal. In the context of this paper, we consider the topological relationship touch as a non- intersecting spatial relationship, or in other words a weak contact that implies a discontinuity as defined in (Asher and Vieu 1995). Such a topological notion, although not very often used in spatial reasoning, is of particular interest for many physical systems. 3. HIERARCHICAL TEMPORAL REASONING Let us analyse the relationships between two events e1(a11, a21, ..., ap1) and e2(a12, a22, ..., ap2) defined using a same calendar C(A1, A2, ..., Ap, ..., Am) and a same granularity m, 1pm. The possible hierarchical temporal relationships that apply between these two events are given by the combination of · Allen's 13 temporal relationships at the granularity level Ap of the these events, · with the following Allen's 5 temporal relationships (before, after, equal, meets, met_by) for each successive higher level of the calendar hierarchy. These 5 temporal relationships reflect the nested structure of the calendar hierarchy, that is, temporal relationships between non-intersecting intervals. As applied to the topological relationship touch for the spatial dimension, meets and met_by are considered here as non-intersecting temporal relationships. Overall, the total number of possible hierarchical temporal relationships, denoted N ht(p), between two events e1 and e2, defined at the same level of granularity p of a calendar C(A1, A2, ..., Ap, ..., Am) is N ht(p) = 13 x ( 5p-1) For example, the possible hierarchical temporal relationships for two events represented with a calendar defined with three levels of granularity C1(Year, Month, Day) are given by the combination of the (before after, equal meets met_by), (before after equal meets met_by) and (before after equal meets met_by starts started finishes finished overlaps overlapped during contains) disjunction of temporal operations, that is N ht(3) = 325 hierarchical temporal relationships. Let us illustrate the potential of our model with a simplified example. We consider a daily crime event that took place between 10:00 and 11:00 on July the 10th (without a loss of generality, we consider an application that takes place during the same year). Crime 5

investigators want to identify closely related events in the same day, and at previous days (we will not consider the spatial dimension in this section). This event is represented using a calendar C1(Month, Day, Hour), its temporal value is e1(July, 10, [10:00,11:00]). Then, events that happen the same year, day and time are evaluated with a (e1 first operand) ht(equal, equal, equal) hierarchical temporal operation

Events that happen the same month and day immediately before or after the time of this event are evaluated with a ht(equal, equal, meet met_by) disjunction of hierarchical temporal operations;

Events that happen the same month, the previous day, and at the same time of this event are evaluated with a ht(equal, met_by, equal) hierarchical temporal operation;

Events that happen the same month, previous non immediate days, and at the same time of this event are evaluated with a ht(equal, after, equal) hierarchical temporal operation;

Events that happen the same month, the previous day, and just before of after the time of this event are evaluated with a ht(equal, met_by, meet met_by) disjunction of hierarchical temporal operations;

Finally, events that happen the same month, previous non immediate days just before or after the time of this event are evaluated with a ht(equal, after, meet met_by) disjunction of hierarchical temporal operations.

Event patterns can be also analysed. Events that happen the same month are evaluated with a ht(equal, t-any, t-any) operation, a same day with a ht(t-any, equal, t-any) operation, at a same time with a ht(t-any, t-any, equal) operation where t-any denotes the possible

temporal operations for a considered level of the calendar hierarchy. These examples show the flexibility of such a hierarchical reasoning approach. The complexity of our model is

relatively limited as calendars are usually defined with a limited number of levels. This

includes, for example, events defined with the finest granularity of a minute or second. For

events with a periodic nature at the day level, investigation at the coarser level of month

and year becomes redundant or less meaningful. Figure 4 illustrates possible combinations with a calendar, defined with three hierarchical levels (inverse relationships are not represented).

Month

Day

Hour

Figure 4: Hierarchical temporal relationships Example of a three level calendar 6

4. HIERARCHICAL SPATIAL AND TEMPORAL REASONING Each level of a spatial hierarchy has its own set of regions and topological relationships that can be used in spatial reasoning. The underlying constraints of the spatial hierarchy imply that two events located in space, respectively represented by two regions that are equal at a level of the spatial hierarchy, are either equal, disjoint or touch at a lower level if any - of the spatial hierarchy. Similarly, two events located in space, represented by regions that touch at a considered level of the spatial hierarchy are either disjoint or touch at a lower level if any - of the hierarchy. Finally, two events located in space, disjoint at a considered level of the spatial hierarchy, are disjoint at a lower level if any - of the spatial hierarchy (Figure 5). These lead to 3 possible hierarchical relationships for a onelevel spatial hierarchy, 6 for a two-level spatial hierarchy, 10 for a three-level hierarchy etc. Overall, the number of possible hierarchical topological relationships, denoted N hs(q), between two events e1(a11, a21, ..., ap1) and e2(a12, a22, ..., ap2), defined at the same level of granularity p of a calendar C(A1, A2, ..., Ap, ..., Am), respectively located in es1 (s11, s21, ..., sq1) and es2 (s12, s22, ..., sq2) at the same level q of abstraction of a spatial hierarchy SH( S1 , S2 , ... , S q , ... , Sn ) , is N hs(q) = 1 + 2 + ... + q + (q +1) We can remark that these hierarchical topological relationships have some singularities if compared to the previously defined hierarchical temporal relationships. The main differences come from the fact that time is oriented whereas space is not; then two additional inverses are defined in time, that is, met_by and after for the temporal relationships meets and before, respectively. Moreover space is not cyclic by nature although time is (i.e., the assumptions of our model). We have also restricted our model to non-intersecting regions at all levels of the spatial hierarchy. Let us remark that a relaxed constraint at the lower level of the spatial hierarchy, that is, application of the complete set of 8 topological relationships as defined in (Egenhofer, 1991) will lead to an increased number of potential topological relationships, that is, 8 x ( 5q-1). In accordance with our hierarchical approach to temporal reasoning, let us assume that the crime event introduced in Section 3 can be located at either the building, neighbourhood or district spatial hierarchy levels. Crime investigators at the neighbourhood level may be interested in those events which happened in the same neighbourhood, whilst investigators working at district level may be interested in those events which happened in the same district. In the spatial dimension, our example event is represented using a spatial hierarchy, its value is SH1(district1, neighb1, building1). Therefore, events that happen within the same district, neighbourhood, and building are evaluated with a hs(equal, equal, equal) hierarchical topological operation; Events that happen within the same district and neighbourhood, but in spatially disjoint buildings are evaluated with a hs(equal, equal, disjoint) hierarchical topological operation; Events that happen within the same district, but in disjoint neighbourhoods are evaluated with a hs(equal, disjoint, disjoint) hierarchical topological operation; 7

Events that happen within the same district and touching neighbourhoods are evaluated with a hs(equal, touch, disjoint touch) disjunction of hierarchical topological operations.

Similarly, Figure 5 introduces topological relationships within a three-level spatial hierarchy in an urban context, say, SH1(district, neighbourhood, building).

District

Neighbourhood

Building

Figure 5: Hierarchical spatial relationships Example of a three level hierarchy Hierarchical relationships have been identified in the temporal and spatial dimensions. Hierarchical relationships can be combined for events located in space. A general rule can be formulated as follows under the constraints of our model defined in the temporal and spatial dimensions: · Let us consider two events e1(a11, a21, ..., ap1) and e2(a12, a22, ..., ap2), defined at the same level of granularity p of a calendar C(A1, A2, ..., Ap, ..., Am), respectively located in es1 (s11, s21, ..., sq1) and es2 (s12, s22, ..., sq2) at the same level q of abstraction of a spatial hierarchy SH( S1 , S2 , ... , S q , ... , Sn ) . The number of possible hierarchical relationships, denoted N h(p,q), between the events e1 and e2 is given by the product of N ht(p) by N hs(q), that is N h(p,q) = (13 x ( 5p-1) ) Ч (1 + 2 + ... + q + (q +1)) Spatio-temporal queries can be then expressed by the conjunction of hierarchical topological and temporal expressions. For example events that happen the same month, previous non immediate days just before or after the time of a crime; in the same district and neighbourhood, but at topologically disjoint buildings as a considered crime event are evaluated with a ht(equal, after, meet met_by) hs(equal, equal, disjoint) expression; Events that happen the same month, the previous day at the same time, in the same district and same or "touching" neighbourhoods as a considered crime event are evaluated with a ht(equal, met_by, equal) hs(equal, touch equal, s-any) expression. As topological and temporal hierarchical operations are orthogonal (i.e., defined independently), previously defined topological and temporal hierarchical queries can be combined in either manner. The proposed language uses a set of well known operations in both temporal and spatial dimensions, so its understanding and appropriation by final users is quite a straightforward task. The large number of resulting combined operations offers a powerful set of operations to support the analysis of patterns in either the temporal, spatial or spatio-temporal domain. Moreover, clustered patterns in space and/or time and periodic 8

events can be studied. Such spatio-temporal reasoning capabilities are particularly adapted to explorative analysis developed in crime or epidemiological studies. It is generally considered that passing and browsing through distinct levels of abstraction imply the composition of represented events (Badaloni and Berati 1994). In particular, real-world events represented at a temporal level of abstraction can only be described at a finer temporal granularity if we also shift to a finer temporal level in order to observe the changes they may produce in the environment (Mota and Robertson 1996). Accordingly, we may consider, in the context of our example, that the analysis of crime events at the city level corresponds to a temporal level of a month, the district level to a temporal level of a day, and neighbourhood and building levels to a temporal level of an hour. Therefore, the analysis of topological and temporal relationships can be represented in a co-ordinated way. For example, one may consider that the city level of the spatial hierarchy can be associated to a temporal granularity of a month, the district and neighbourhood spatial levels of abstraction to a temporal granularity of a day, and finally the building spatial level to a temporal granularity of an hour; obviously, these assumptions are application dependent. 5. CONCLUSION The analysis of spatio-temporal phenomena in GIS often implies reasoning at different temporal and spatial levels of abstraction. The research described in this paper introduces an integrated approach to the development of hierarchical reasoning in time and space. The model is based on the manipulation of events represented as hierarchical data types modelled using the concept of calendar that reflects the cyclic nature of time. Resulting temporal operations are then identified as the possible combinations of temporal operations at each level of the temporal hierarchy. By extension, we apply similar hierarchical concepts to the spatial dimension. This leads to a multi-scale and hierarchical representation of event relationships in space and time. Overall, the model supports the identification of relationships in time and space at different hierarchical levels. The model is flexible enough to support application orientated relationships in both temporal and spatial dimensions. We have illustrated this flexibility with events "contained" within a period unit defined at the immediate superior level of their calendar hierarchy and located in non-intersecting regions. Similar or relaxed - constraints can be defined according to application dependent strategies. This hierarchical approach of relationships in space and time offers a flexible algebra for the manipulation of geo-referenced events. As our model is based on well-known relationships in time and space, its computational complexity is relatively limited, and its implementation is feasible. It is flexible enough to apply in different application contexts such as crime studies, urban traffic monitoring or epidemiological studies. Current work concerns the computational implementation of these hierarchical relationships and the extension of the model to events defined with intersecting space-time hierarchical units. REFERENCES Allen, J. F. (1984). "Towards a general theory of actions and time", Artificial Intelligence, 23, 123-154. Allen, J. F. (1991). "Temporal reasoning and planning", in Reasoning About Plans, Morgan-Kaufman, SanMateo, CA, 1-68. 9

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