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Spaccapietra, 2006; Jensen et al., 2004; Pourabbas,
2001; Sampaio et al, 2006, Silva et al., 2008).
Ahmed & Miquel (2005) define a spatial
dimension according to the raster model as an
infinite set of members. They simulate this “infi-
nite” set using interpolation functions to calculate
measures in every point of the analysis region
represented by spatial dimensions. Pourrubas
(2003) provides a formal framework to integrate
multidimensional and geographic databases. The
model handles spatial dimensions in which mem-
bers of different levels are related by topological
inclusion relationship. These models only handle
numerical SQL aggregation functions. Sampaio et
al. (2006) define a logical model for spatial data
warehouse using the object-oriented approach.
Spatial measures and numerical measures can be
aggregated using SQL spatial and numerical ag-
gregation functions. The model does not provide
any support for correct aggregation. Damiani &
Spaccapietra (2006) define a model in which
all levels of a spatial dimension can be used as
measures, allowing multi-granular analysis. The
model aims at supporting measures as geographic
objects, but it does not explicitly represent spatial
and alphanumeric attributes. Then, no aggregation
constraint is defined on measures (attributes of
geographic objects).
Silva et al., (2008) define formally the ele-
ments of a spatial data warehouse with particular
attention to aggregation functions for spatial data.
They introduce a set of aggregation functions that
combine numerical and spatial functions. These
aggregation functions are classified according to
whether numerical aggregation is scalar, distribu-
tive or holistic, and whether spatial aggregation
is unary or n-ary function. However, these ag-
gregation functions are not associated with any
aggregation constraints and numerical functions
are applied exclusively to metric values of spatial
data (i.e. perimeter, etc.).
Only, Jensen et al., (2004) and Pedersen et al.,
(2001) define logical multidimensional models
for spatial data warehouses taking into account
aggregation constraints. In particular, Jensen et
al., (2004) propose a model for location-based
services. It supports partial inclusion of spatial
dimension members granting imprecise measures.
Measures are numerical and alphanumeric values.
The model extends (Pedersen et al., 2001) and it
grants correct aggregation of non-spatial measures
by classifying measures according valid aggrega-
tion functions (c.f. the Section “Multidimensional
Models”). Nevertheless, the model does not ex-
plicitly represent spatial measures nor associated
aggregation constraints.
Pedersen & Tryfona (2001) investigate pre-
aggregation in multidimensional applications
with spatial measures. In particular, they study
the pre-aggregation of alphanumeric attributes
associated with spatial measures. In other words,
they investigate the correct aggregation of alpha-
numeric attributes of geographic objects according
to their spatial components. The model represents
measures as bottom levels of dimensions. Ag-
gregated measures are less detailed dimensions
levels. Since a topological inclusion relationship
always exists between spatial objects at different
levels, the model does not allow applying spatial
disaggregation functions. Then, only union can
be used. Spatial disaggregation functions, such as
intersection or centroid, can not be applied.
To conclude, few SOLAP models consider
aggregation constraints on spatial and numerical
measures without correctly support aggregation
of geographic measures. Indeed, as shown in the
next section, correct aggregation of geographic
measures should take into account: semantics
of measures, used dimensions, overlapping ge-
ometries and dependency between spatial and
alphanumeric functions.
reSeArch MotIvAtIonS
The aggregation of geographic measures raises
several problems from both theoretical and
implementation points of view. In particular,
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