Relative adjacencies in spatial pseudo-partitions Roderic Béra 1, Christophe Claramunt 1 1 Naval Academy Research Institute, Lanvéoc-Poulmic, BP 600, 29240 Brest Naval, France {bera, claramunt}@ecole-navale.fr Extended abstract Introduction Qualitative spatial reasoning is a valid support for inferring relationships in geographical spaces. In particular, qualitative inferences are cognitively more expressive than quantitative ones, and support reasoning mechanisms in the absence of complete spatial knowledge (Cohn 1997). In geographical space, reasoning on spatial entities is supported by representations that mainly involve topological (Pullar and Egenhofer 1988, Egenhofer 1991, Randell et al. 1992, Clementini et al. 1993, Cui et al.1993) and direction relationships (Freksa 1992, Frank 1996, Sharma 1996, Papadias and Egenhofer 1997, Goyal and Egenhofer, 2000). Those spatial relationships provide useful mechanisms to evaluate the mutual relationships of regions over space. An effective data structure for modelling geographical spaces is based on a partition of space, thus forming a coverage derived from thematic classification (Robinson et al. 1984, Frank et al. 1997). Topological relationships in spatial partitions are mainly based on two operators: adjacency and disjunction. But these operators do not provide many capabilities to qualify the relationships between the regions forming a spatial partition. Alternatives include spatial statistic operators that estimate the spatial variability of a property but not the way elements of this distribution are interrelated. We believe that in many situations there is an interest in analysing how two given regions are in relation one with respect to the other. This should also help to identify local and global structural patterns in a spatial partition. Modelling principles To take advantage of some of the dual graph properties of spatial partitions and to introduce a generalisation of the adjacency relationship, we introduce and give a formal definition of a relative adjacency operator that evaluates to which degree regions in a given spatial partition are mutually integrated in the dual graph derived from adjacency relationships. The operator is flexible enough to evaluate those relative adjacencies at different levels of magnitude, that is, by minimising versus maximizing the impact of outlying regions. Given a reference region in a spatial partition we also show how the relative adjacency operator can support the analysis of the relative distribution of other regions, and how those regions are clustered with respect to that reference region.
COSIT 03 Doctoral Colloquium We briefly discuss the differences that one can observe in analysing how near and distant regions are interrelated. First, let us consider the spatial partition derived from American countries. With respect to an adjacency relationship, it appears that Uruguay is more integrated with Brazil than the reverse. This reflects the computational property that the probability of moving randomly one step from Uruguay in the dual graph of the American countries and reaching Brazil is higher than the contrary. Secondly, one should also say that if near things are related (Tobler 1970), distant things, although less related, are related too and in different ways that reflect their integration versus segregation in the dual graph. This also corresponds to the essential intuition that some of the spatial relationships identified in a spatial partition are not always symmetric (Duckham and Worboys, 2001). In order to develop further our modelling approach we introduce some basic properties and definitions of topological spaces. Let C be a topological space, and let x be a region of C, that is a connected subset of C. x o denotes the interior of x and x its boundary according to the usual notations. The mathematical definition of a partition implies that the elements of the partition don t intersect. However this doesn t fit very well geographical spaces as adjacent regions share part of their boundaries. Therefore, we propose a slightly relaxed definition of the partition where the union of its elements still gives the set but where the interior of the elements don t intersect. Let us introduce the topological relationship touch, expressing that for two regions x 1 x 2 = x 1, x 2 of C, x 1 touches x 2 iff x1 x2 1iff x The adjacency operator is then defined as Adj(x 1, x 2 ) = 1 touches x1 0 otherwise The adjacency set of a region x of C is the union of the regions of C adjacent to x, i.e. Adj(x) = {x i C / Adj(x,x i ) = 1} We denote CAdj(x) the cardinality of Adj(x). The dual graph G of a pseudo-partition X pp is given by the pair (E,N), where N is the set of regions of X pp and E the set of edges subset of the Cartesian product N N, where (x i,x j ) E iff Adj(x i,x j ) = 1. The relative adjacency R(x p,x r ) of two regions x p and x r of X pp is then defined as follows: R( xi, xr ) R( x p, xr ) = d Adj( x p, xr ) + (1 d) (1) CAdj( x ) x Adj( i x p where 0 < d < 1 By convention we say that the higher R(x p, x r ), the higher x p s integration with respect to x r (vs. the lower their level of segregation). The relative adjacency between two regions is defined recursively. It takes into account the importance of the adjacency between the two given regions x p and x r (first term) and to which degree the k-neighbourhoods of x p are relatively adjacent to x r (second term, where a k-neighbourhood of x p is defined as a region which is k-step away in the dual graph G). The damping coefficient d is user-defined and balances the relative importance of adjacencies and relative adjacencies. Higher values of d lead to a smaller ) p
account of outlying regions, whereas smaller ones minimise the importance of closer regions. The relative adjacency operator is not symmetric, except in some special cases where symmetry is caused by a symmetrical configuration of the spatial pseudopartition. Relative adjacency values are non null except for regions disconnected in the dual graph G. The relative adjacency is drawn by the unit interval as it is made of a sum of adjacency values (i.e. 0s and 1s) weighed by their adjacency number and the multiplicative coefficient d. Application to a case study In order to illustrate how some structural patterns can be observed with the relative adjacency operator, we introduce the large-scale pseudo-partition example of the countries of America. The dual graph (figure 1) has a high diversity of node degrees, a highly connected side (Southern America) as well as a poorly connected peripheral side (central and northern part of the continent). Can USA Mex Bel Gua Sal Hon Nic Cos Pan Ven Guy Sur Equ Col Bra Fgu Per Bol Par Uru Fig. 1. Dual graph for the pseudo-partition of America s countries A first level of analysis is produced by the observation of the quantitative values given by the relative adjacency for a user-defined value of the damping coefficient d. In order to extend the dimension of the relative adjacency values we observe the evolution of relative adjacency values as a function of d. Figure 2 shows the following patterns for the integration of American countries to Brazil. The curve (I) is given by the self-relative adjacency, cluster (II) regroups Brazil s adjacent countries, (III) corresponds to the most remote countries (North and Central America), (VI) gives Panama which is adjacent to Colombia, one of the Brazil s adjacent countries. But as Panama is not very much integrated with the graph, its curve differs slightly from those obtained for the other 2-neighbours (curve V, Equator and IV, Chile) which are more connected to the 1-neighbours of Brazil. Chi Arg
COSIT 03 Doctoral Colloquium 1 Brazil 0.9 0.8 0.7 II 0.6 0.5 IV I V 0.4 0.3 0.2 III VI 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d Fig. 2. Brazil s relative adjacencies Conclusion The relative adjacency operator qualifies degrees of mutual integration in pseudopartitions, together with a damping coefficient that outlines either neighbouring or outlying regions. Such a coefficient and observed trends, derived from its computation, can even be modulated in function of the context and properties intrinsic to the underlying spatial pseudo-partition and the phenomenon represented. Although the relative adjacency measures a mutual integration factor for some given spatial regions at the local level, it can be also considered as a form of qualitative distance, and a mean to analyse the structure of a given spatial partition. Relative adjacencies can be also moderated with respect to additional spatial and aspatial properties that will enrich the approach. For instance, region sizes and length of the boundaries shared by two adjacent regions, distances between centroids are relevant parameters to consider. Further work concerns experimental validations of the approach to evaluate to which degree relative adjacencies correlate cognitive interpretations. References Clementini, E., Di Felice, P. and Van Oosterom, O., 1993, A small set of topological relationships suitable for end-user interaction. In: D. J. Abel and B. C. Ooi (eds.). Advances in Spatial Databases. Springer-Verlag, Singapore, pp. 277-295. Cohn, A. G., 1997, Qualitative spatial representation and reasoning techniques. In G. Brewka, C. Habel and B. Nebel (eds.), Proceedings of KI-97, Springer-Verlag, LNAI 1303, Berlin, pp 1-30.
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