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on the x-axis represents a class of schema matchings with a different precision. The
z
-axis represents the similarity measure. Finally, the y-axis stands for the number of
schema matchings from a given precision class and with a given similarity measure.
Two main insights are available from Fig.
3.2
. First, the similarity measures of
matchings within each schema matching class form a “bell” shape, centered around
a specific similarity measure. Such a behavior indicates a certain level of robust-
ness of a schema matcher, assigning close similarity measures to matchings within
each class. Second, the “tails” of the bell shapes overlap. Therefore, a schema
matching from a class of a lower precision may receive a higher similarity mea-
sure than a matching from a class of a higher precision. This, of course, contradicts
the monotonicity definition. However, the first observation serves as a motivation
for a definition of a statistical monotonicity, first introduced in
Galetal.
[
2005a
], as
follows:
Let ˙
Df
1
;
2
;:::;
m
g
be a set of matchings over schemata S
1
and S
2
with n
1
and n
2
attributes, respectively, and define n
D
max.n
1
;n
2
/.Let˙
1
;˙
2
;:::;˙
n
C
1
iff
i
1
n
p./ <
n
be subsets of ˙ such that for all 1
.
We d e fi n e M
i
to be a random variable, representing the similarity measure of a
randomly chosen matching from ˙
i
. ˙ is
statistically monotonic
if the following
inequality holds for any 1
i<j
n
C
1:
i
n
C
1;
2
˙
i
˝.M
i
/<
˝.M
j
/;
where ˝.M/ stands for the expected value of M .
Intuitively, a schema matching algorithm is statistically monotonic with respect
to given two schemata if the expected certainty level increases with precision.
Statistical monotonicity can assist us in explaining certain phenomena in schema
matching (e.g., why schema matcher ensembles work well [
Gal and Sagi 2010
])
and also to serve as a guideline in finding better ways to use schema matching
algorithms.
3
Contextual Attribute Correspondences
Attribute correspondences may hold under certain instance conditions. With con-
textual attribute correspondences, selection conditions are associated with attribute
correspondences. Therefore, a contextual attribute correspondence is a triplet of
the form .A
i
;A
j
;c/,whereA
i
and A
j
are attributes and c is a condition whose
structure is defined in Sect.
3.1
.
Example 2.
With contextual attribute correspondences, we could state that
R.Card-
Info.cardNum
is the same as
S.HotelCardInfo.clientNum
if
R.CardInfo.type
is assigned with the value
RoomsRUs
. For all other
type
values,
R.Card-
Info.cardNum
is the same as
S.CardInfo.cardNum
. These contextual attribute
correspondences are given as follows.
