Quantitative methods in marketing 229
conditional probabilities of allocation at least
equal to . More formally:
-positiveregion of the set ZU:POSP(Z)=
Pr (ZXi)≥,{XiE(P)} with PC.Following An et al. (1996), is defined to lie
between 0.5 and 1. Hence for the current
example, the condition equivalence class X 1 =
{o 1 , o 4 ,o 6 } have a majority inclusion (with at
least 60 per cent majority needed, i.e. = 0.6) in
YH, in that most objects (two out of three) in X 1
belong in YH. Hence X 1 is in POS0.6C (YH). It
followsPOS0.6C (YH)={o 1 ,o 4 ,o 5 ,o 6 ,o 7 }.
Corresponding expressions for the
-boundary and -negative regions are given
by Ziarko (1993a) as follows:
-boundaryregion of the setZU:BNDP(Z)=
1–〈Pr(ZXi)〈β{XiE(P)} with PC,-negativeregion of the setZU:NEGP(Z)=
Pr (ZXi)≤1–,{XiE(P)} with PC.UsingPandZfrom the previous example, with
= 0.6, then BND0.6C (YH) = 0 (empty set) and
NEG0.6C (YH) = {o 2 ,o 3 }. Similarly, for the decision
class YLit follows that POS0.6C (YL) {o 2 , o 3 },
BND0.6C (YL) = 0 and NEG0.6C (YL)={o 1 ,o 4 ,o 5 ,o 6 ,
o 7 }.
VPRS applies these concepts by firstly seeking
subsets of the attributes, which are capable (via
construction of decision rules) of explaining
allocations given by the whole set of condition
attributes. These subsets of attributes are
termed-reductsorapproximate reducts.Ziarko
(1993a) states that a -reduct, a subset Pof the
set of conditional attributes Cwith respect to a
set of decision attributes D, must satisfy the
following conditions: (i) that the subset Poffers
the same quality of classification (subject to the
samevalue) as the whole set of condition
attributesC; and (ii) that no attribute can be
eliminated from the subset Pwithout affecting
the quality of the classification (subject to the
samevalue).
The quality of the classification is defined
as the proportion of the objects made up of the
union of the -positive regions of all the
decision equivalence classes based on the con-
dition equivalence classes for a subset Pof the
condition attributes C.
As with decision tree techniques, ceteris
paribus, a clear benefit to users of VPRS is
the ability to interpret individual rules in a
decision-making context (as opposed to inter-
preting coefficients in conventional statistical
models). Hence VPRS-generated rules are rela-
tively simple, comprehensible and are directly
interpretable with reference to the decision
domain. For example, users are not required to
possess the technical knowledge and expertise
associated with interpreting classical models.
These VPRS characteristics are particularly
useful to decision makers, who are interested in
interpreting the rules (based on factual cases)
with direct reference to the outcomes they are
familiar with.Dempster–Shafer theory
The Dempster–Shafer theory (DST) of evidence
originated in the work of Dempster (1967) on
the theory of probabilities with upper and
lower bounds. It has since been extended by
numerous authors and popularized, but only to
a degree, in the literature on artificial intelli-
gence (AI) and expert systems, as a technique
for modelling reasoning under uncertainty. In
this respect, it can be seen to offer numerous
advantages over the more ‘traditional’ methods
of statistics and Bayesian decision theory. Hajek
(1994) remarked that real, practical applications
of DST methods have been rare, but subsequent
to these remarks there has been a marked
increase in the applications incorporating the
use of DST. Although DST is not in widespread
use, it has been applied with some success to
such topics as face recognition (Ip and Ng,