NON-ADDICTIVE SAMPLE BASED FACTS - Deetc

attribute keys and K is the remaining attribute. From
an OLAP point of view, X1, X2, ..., Xn turn to be the
axis of the cube and each value of attribute K is in
the intersection of those axis. We can express K as a
function
K : f (X1,X2,...,Xn) (5)
If one of the axis is omitted from (5), e.g Xn, we
are performing a similar operation as a dimensional
reduction
Knew : f (X1,X2,...,Xn−1) (6)
However, in (5), the result set does not contain
duplicates, because the key set is contained in
the projection result. In (6), to guarantee the
distinction, K needs to be aggregated for each
distinct tuple of the projection set PS set defined as
(x1,x2,x3,...,xn)|xn ∈ dom(Xn). Using the sum as the
aggregation function, the attribute K is summarised as
∑
xn∈dom(Xn)
f (x1,x2,...,xn−1,xn) (7)
The facts in the OLAP Cube does not always directly
point to an a single attribute. Their definition can be
based upon a mathematical operation applied to one
or more data elements. These are generally referred
to as derived data or derived facts [5]. If the operation
defines a ratio3 , the application of (7) results in a nonmeaningful
value, because ratios are non-addictive.
The performance indicators cannot be pre-calculated
and stored directly in the fact table; it must be implemented
as a derived fact.
To be able to properly calculate the performance
indicators for the domain, e.g. rat, a corrective
weight must be used, as can be see in (4). Since
query’s targets are user depended, determined in
runtime by the users’ restrictions over the Sociodemographic
dimension, one can’t pre-determine the
right weight to use. It’s necessary to store all the
possible weights, using (3) to calculate them. For a 4
attribute key cube, we are dealing with a theoretical
value of 15 possible combinations, sum 4 k=1 (4 i
). If one
decided to stored all the possible weights as facts, for
each tuple of the fact table 14 unnecessary values are
stored, as only one is valid for each query. Besides,
the number of possible targets increase this number.
This approach is also not feasible because, for some
indicators, it is necessary to store the weights for
non-viewers, that is, individuals that didn’t watch
television during the analysis period. Fact tables store
only one type of occurrence, in this case, the fact that
some individual watched television; it’s not a good
3 Ratios are not the only type of non-addictive facts.
practice to store the opposite fact too. One must use
a more straightforward approach, using some domain
knowledge.
The data are quota sample-based, which means the
sample was designed to be a representative subset of
the population, regarding some descriptive characteristics.
In this sense, it’s impossible to have an individual
to support simultaneously two or more values for
an attribute. For example, an individual that is present
in the target males with ages ranging from 4 to 14 cannot
be part of other target, females with ages ranging
from 4 to 14.
Property 1. Let Ta be a target with a restriction over
a socio-demographic attribute a,
ta=v1 ∩ta=v2 = /0
If the weights are normalised, which can be done during
the ETL 4 proccess, for any two or more disjoint
subsets, their weights sum up to one.
Property 2. Let Tb be a target with a restriction
over socio-demographic two valued attribute b,
∑ Ta=0∈PW(i) + ∑ Ta=1∈PW(i) = 1 is always true.
Knowing that, it’s possible to create a fact table that
store the weights for all possible targets, including
the non-viewers individuals. That fact table shares
the data and socio-demographic dimensions. Each tuple
in the fact table represent the reference value for
a specific combination of socio-demographic values,
for a given day, applying (3) for each distinct combination.
Figure 2 illustrate the Contact star-schema.
Figure 2: Illustration of the contact starschema model.
The former model provide the weights for all the contacted
individuals, viewers or not, for one day. To calculate
the performance indicators for the domain, e.g.
rating, it’s also necessary to determine the weights of
the viewers. To address this issue, it’s necessary to
create another fact table, that store the viewers’ daily
corrective weight. Since this table is indexed by all
of the dimensions discussed so far, Date, Time, Program
and Socio-Demographic, a tuple represents
4 Extraction Transformation and Loading