Spatial Data Warehouses - Data Warehouse Systems: Design and Implementation - page 425

￿ Integral : S f ( x, y ) dx dy .

￿ Area : S dx dy .

1+ ∂ ∂x 2 + ∂ ∂y 2 dx dy .

￿ Surface : S

From these operations, other derived operations can be defined. These are
prefixedwithan' F ' (field) in order to distinguish them from the usual
aggregation operations generalized to fields, which we discuss below.

￿ FAvg : Integral / Area .

￿ FVariance : S ( f ( x,y ) − FAvg ) 2 dx dy .

￿ FStDev : √ FVariance .
Finally, FMin and FMax return, respectively, the minimum and maximum
value taken by the function. These are obtained by Min(RangeValues(

Area

· )) and

Max(RangeValues( · )), where Min and Max are the classic operations over

numeric values.
All operations on basic types are generalized for field types. This is called
lifting .Anoperation op for basic types is lifted to allow any of the arguments
to be replaced by a field type and returns a field type. As an example, the less
than ( < ) operation has lifted versions where one or both of its arguments can
be field types and the result is a Boolean field type. Intuitively, the semantics
of such lifted operations is that the result is computed at each point in space
using the nonlifted operation. For example, applying the lifted ' < 'operation
to two fields that describe the temperature at 2 days will result in a Boolean
field that states at each point whether the temperature of the first field is
smaller than the one in the second. When two fields are defined on different
spatial extents, the result of a lifted operation is defined in the intersection
of both extents and undefined elsewhere.
Aggregation operations are also lifted. For instance, a lifted Avg operation
combines several fields, yielding a new field where the average is computed
at each point in space. Lifted aggregation operations are used in particular
for granularity transformations. For example, a lifted average could be used
to transform a temperature field of granularity day to granularity month.

11.2 Conceptual Modeling of Spatial Data
Warehouses

In this section, we explain the spatial extension of the MultiDim model. For
this, we use as example the GeoNorthwind data warehouse, which is the
Northwind data warehouse extended with spatial types. As shown in the
schema in Fig. 11.4 , pictograms are used to represent spatial information.