A Multidimensional Model for Correct Aggregation of Geographic Measures - Evolving Application Domains of Data Warehousing and Mining - page 175

graphic measures and the dependency of spatial
and alphanumeric functions, and then we introduce
the constraints.

Definition. (DimensionsMeasure).
For each Entity Schema S e and set of levels
of a View V e , we assume a function Dimensions-
Measure:A(S e )× L(V e ) → {AF ∑ ,AF ω ,AF c } which
returns the aggregation type.
Example 5. In our example, SemanticsMeasu
re(nbDamagedTrees) = AF ∑ , because the sum of
number of trees makes sense, and DimensionsM
easure(nbDamagedTrees, S year , S phenomenon )= AF ω ,
because number of trees is not additive on the
time dimension (the level S year, which is not the
most detailed level of the “Time” dimension, is
used in the multidimensional query).

Semantics of Alphanumeric Attributes
of Geographic Measures

Following the approach to ensure correct aggrega-
tion, described in the Section “Multidimensional
Models”, we introduce the functions Dimensions-
Measure and SemanticsMeasure .
They classify aggregation functions of at-
tributes of geographic objects in three different
groups, called aggregation types : AF ∑ , AF ω and
AF c , where AF ∑ > AF ω > AF c , “>” being a total
order. Functions classified as AF ∑ can be applied
to measures that can be summed, AF ω are func-
tions that can be applied to measures that can be
averaged, andAF c are functions that can be applied
to measures that can be only counted.
SemanticsMeasure takes into account the
semantics of attributes. It takes as input an al-
phanumeric attribute and returns an aggregation
type (AF ∑ , AF ω , AF c ). For example, functions that
can be applied to the number of trees are AF ∑ , as
number of trees can be summed.

Dependency of Spatial and
Alphanumeric Functions

In order to model dependency of spatial and
alphanumeric functions, we introduce a function
SpatialType, which takes as inputs the type of
spatial function (AF U , AF Ω ) and the semantics of
the alphanumeric attribute ( SemanticsMeasure ),
and returns the aggregation type (AF ∑ , AF ω , or
AF c ) that can safely be applied to the alphanumeric
attribute. Let us note spatial aggregation functions
as AF U , and spatial disaggregation functions as
AF Ω . Then, spatial functions classified as AF U al-
lows summing additive data (e.g. number of trees)
because they preserve the set of input geometries
in the result (i.e. union, convex hull, etc.). AF Ω do
not allow summing data as they leave out some
spatial objects (i.e. centroid, intersection, etc.).

Definition. (SemanticsMeasure)
For each Entity Schema S e , we assume a func-
tion SemanticsType: A(S e ) → {AF ∑ , AF ω , AF c }
which returns the aggregation type.
□
DimensionsMeasure takes into account the
semantics of attributes, and used dimensions
levels. It takes as input an alphanumeric attribute
and the dimensions levels of the View , and it
returns an aggregation type (AF ∑ , AF ω or AF c ).
For example, functions that can be applied to the
number of trees aggregating on the time dimen-
sion are AF ω , as the number of trees cannot be
summed on the time dimension (some trees must
not be counted twice.

Definition. (SpatialType function).
We define a function SpatialType: { AF U , AF Ω }
× { AF ∑ , AF ω , AF c } → { AF ∑ , AF ω , AF c } as de-
fined inTable 2.
The function SpatialType models the depen-
dency of spatial and alphanumeric functions
as defined by the Step 3 of our algorithm. In
particular, when using spatial aggregations
(AF U ), it is possible to sum additive alphanu-
meric attributes (AF ∑ ) values of geographic