A Multidimensional Model for Correct Aggregation of Geographic Measures - Evolving Application Domains of Data Warehousing and Mining - page 167

used dimensions). Some logical multidimensional
models define explicitly the type of aggregation
functions that can be applied to measures (Abello
et al., 2006; Lenher, 1998; Pedersen et al., 2001;
Trujillo et al., 2000). These models, based on
the definitions introduced by Rafanelli & Ricci
(1983), classify measures according to three dif-
ferent types of aggregation functions that can be
applied to them: ∑ (data can be added together,
e.g. population), ω (data can be used for average
calculations, e.g. temperature) or c (constant data
implying no application of aggregation operators,
e.g. name). Considering only the SQL standard
aggregation functions (AF) applying to each type
of data ( c, ω and ∑), an inclusion relationship
exists so that AF c = {COUNT} ⊂ AF ω = {AVG,
MIN, MAX, COUNT} ⊂ AF ∑ = {SUM, AVG,
MIN, MAX, COUNT}. This inclusion relation-
ship allows deducing that “∑ data” which can be
summed can also be averaged and identified as the
minimum or the maximum of a set (as “ω data”)
and that “ω data” can also be counted as “ c data”.
By transitivity, “∑ data” can also be counted.

values, such as the number of inhabitants, can be
aggregated using the SUM operator. For the disag-
gregation process (moving to smaller units), the
sum is forbidden as it makes non-sense (Charre
et al., 1997; Chrisman, 1974; Egenhofer & Frank,
1986). Finally, we underline that the concept of
analysis dimension is not present. Indeed, all the
aggregation rules are defined exclusively on the
spatial dimension, while we claim that performing
analysis on other dimensions (e.g. time dimension)
can reveal itself interesting.

gIS Models

Spatial aggregation operators have been formal-
ized and implemented in Geographic Information
Systems (GIS) (Rigaux & Scholl, 1995; Voisard
& David, 2002; Longley et al., 2001). In all these
approaches, unlike the geostatistic solutions, the
semantics of alphanumeric attributes are not taken
into account. This lack is also evident from the
implementation point of view. Nowadays, com-
mercial GIS systems ( i.e. ArcGIS, MapInfo, etc.)
implement only union and splitting operators.
They provide a simple control on the aggregations
applied to alphanumeric attributes using only
their type ( i.e. numeric, alphanumeric, etc.) but
not their semantics. These tools propose some
numeric aggregation functions ( i.e. sum and
average) for the numeric attributes and some
particular methods, as for example the usage of a
default value or of the count operator, for textual
attributes. They do not consider the semantics
of the attributes. Therefore, it is possible to sum
temperatures or population densities, which
makes no sense. Moreover, all other spatial ag-
gregation operators ( i.e. the convex hull, etc.)
only create new geometries without aggregating
alphanumeric attributes.

geostatistic Models

The aggregation of geographic information is cru-
cial in spatial analysis especially in the geostatistic
domain. Spatial aggregation operators ( i.e. union,
convex hull, etc.) and the management of aggre-
gations applied to alphanumeric attributes have
been widely discussed. Different frameworks,
dealing with the type of alphanumeric attributes
and with their aggregation functions have been
proposed to address the issue of disjoint spatial
units aggregation into bigger spatial units (Charre
et al., 1997; Chrisman, 1974; Chrisman, 1998).
Alphanumeric attributes are grouped into classes
and they are associated with particular aggregation
rules. For example, a rule defines that attributes
representing relative values such as the tempera-
ture or the population density, can be aggregated
using a weighted average. Another rule expresses
that attributes used to represent raw quantitative

Spatial olAP Models

Some SOLAP logical models have been proposed
in literature (Ahmed & Miquel, 2005; Damiani &