A Multidimensional Model for Correct Aggregation of Geographic Measures - Evolving Application Domains of Data Warehousing and Mining - page 166

Figure 2. Facts table and cartographic representation of geographic measures

relAted Work

(i.e. Tao & Papadias, 2005; Rao et al., 2003Zhang
& Tsotras, 2005). These methods speed-up ag-
gregation computation on evolving user-defined
spatial hierarchies. While numerical aggregation
functions have been well defined (i.e. SUM, MIN,
MAX, etc.), a standard set of spatial aggregation
functions for spatial measures (geometries) has
not been defined yet. (Shekar et al., 2001) classify
spatial aggregations as distributive, algebraic, and
holistic in order to grant summarazability (c.f.
Sec “Multidimensional Models”) in spatial data
warehouse. Lopez & Snodgrass (2005), and Silva
et al., (2008) formally define a set of spatial and
numerical aggregation functions (i.e. AvgArea,
Count_AtNorth, etc.). In addition to the conceptual
aspects of spatial aggregation, some efforts have
been done for improving queries computation
in large spatial data warehouses. They provide
indexes materialization, selection of aggregated
measures and computational geometry algorithms
(i.e. Stefanovic et al., 2000; Han et al., 1998).
Finally, some models have been proposed for
spatial data warehouses, which formally define
SOLAP main concepts: spatial dimensions, spa-
tial measures, multidimensional operators and
aggregation functions. In particular, the correct
aggregation of geographic information is a quite
complex task, which raises some unresolved
problems as described in the next sections.

In this Section, we investigate problems and
solutions related to aggregation of geographic
data according to different models: logical mul-
tidimensional models, GIS models, geostatistic
models and logical SOLAP models.

Multidimensional Models

The correct aggregation of measures is of crucial
importance in the multidimensional analysis
process. The correctness of aggregation depends
on both the semantics of the measure and the
multidimensional structure of the data warehouse
(Horner et al., 2004; Pedersen et al., 2001). For
example, the sum of pollution values has no
sense, while the max or the min operators can
safely be applied. Also, it is possible to sum
the populations of cities of a region, but to sum
these values on the time dimension is not correct
(the same inhabitants would be counted several
times). This problem is known in OLAP literature
as Additivity (Kimball, 1996). A measure is: (1)
Additive if the sum operator can be applied on all
dimensions (2) Semi-additive if the sum can be
applied on a subset of dimensions (3) Non additive
if the sum makes no sense whatever the dimen-
sion. The correct aggregation means providing
aggregation constraints (i.e. a control on the type
of the aggregation, considering the semantics of
the measure (i.e. the nature of the measure) and