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ear ,S type ), where DimensionType(nbDamagedTree
s,S year ,S type )= AF ω ).
The Horizontal Aggregation Constraint de-
fines correct functions for φ taking into account the
semantics of the alphanumeric attributes thanks to
the function SemanticsType (e.g.. number of trees
can be summed), and the dependency between
spatial and alphanumeric functions thanks to the
SpatialType function (when using intersection, it
is not possible to sum the number of trees).
For example, in order to aggregate geographic
objects resulting from Vertical Aggregation Step
of figure 3, the Horizontal Aggregation Constraint
does not allow using the sum for number of trees.
Therefore, φ nbDamagedTrees = Weighted Average on
surface because a spatial disaggregation (inter-
section) is used for geometry (Figure 6).Indeed,
sum is not representative of number of trees in
the intersection region (φ nbDamagedTrees AF ω =
SpatialType(AF Ω , Min(SemanticType(nbDama
gedTrees))).
By this way, user chooses the functions φ and
κ, and the model avoids building any non-sense
aggregated geographic measures.
In the following, we formally introduce the
Correct Geographic Aggregation Mode and
then the functions DisjGeoObjects and Overlay-
GeoObject.
t d 1 …,t d m be the instances calculated us-
ing OverlayGeoObjects on input instances
and t d i have the same geometry and coor-
dinates than ti i D1 , … t i Dr of the vertical in-
stances (See Zone V1, Zone V2 and Zone
V3, Figure 4)
SpatialAggregationType be the spatial
function type { AF U , AF Ω }
geomAggregate be the result of the spatial
aggregation on the geometries of t 1 …,t n
The alphanumeric aggregation function ϕ i
of Φ e be defined by means of:
φ i (t d i .a m , … t d i .a l , t d i .geom) (Horizontal
Aggregation)
κ i (t i Di .a m ,…,t i Di .a l ) (Vertical Aggregation)
such as:
1. ϕ i = φ 1
2.
t d i .a i 1 r (t i Di .a m ,…,t i Di .a l )
then the Geographic Aggregation Mode Φ e
is correct if are respected the following con-
straints:
1.
Vertical Aggregation Constraint:
κ i Min(DimensionsMeasure(a m , S l1 ,…
S lm ),…,DimensionsMeasure(a l , S l1 , …, S lm ))
Definition. (Correct Geographic Aggrega-
tion Mode)
Let:
2.
Horizontal Aggregation Constraint:
φ i SpatialType(SpatialAggregation
Type, Min(SemanticsMeasure(a m ),…,
SemanticsMeasure(al)) l ))
V e be a View (See example 4: V zones-year = áB-
C naturalrisks , áS phenomenon , S year ñ, Θ zone , γñ)
t 1 …,t n be instances of the Geographic
Entity which must be aggregated: input
instances (See Zone A and Zone B, Figure
2)
Definition. (DisjGeoObjects)
DisjGeoObjects is a function which takes as
inputs n Geographic Entity Instances and returns
l ≥ n Geographic Entity Instances whose geom-
etries are obtained using the geometric intersec-
tion operator.
Figure 3 shows the results of the DisjGeoOb-
jects function on geographic objects of figure 2.
t i D1 , … t i Dr be the instances calculated using
DisjGeoObjects on input instances: verti-
cal instances (See Zone 1 of A, Zone 2 of
A, Zone 1 of B and Zone 2 of B, Figure 3)
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