Databases Reference
In-Depth Information
The monotonicity principle refers to the performance of complete schema match-
ings from which the performance of individual attribute correspondences can be
derived. Assume that out of the n
n
0
attribute matchings, there are c
n
n
0
cor-
rect attribute matchings, with respect to the exact matching. Also, let t
c be the
number of matchings, out of the correct matchings, that were chosen by the match-
ing algorithm and f
n
n
0
c be the number of incorrect attribute matchings.
Then, precision is computed to be
t
t
C
f
and recall is computed as
c
. Clearly, higher
values of both precision and recall are desired. From now on, we shall focus on the
precision measure, where p./ denotes the precision of a schema matching .
We first create equivalence schema matching classes on 2
S
. Two matchings
0
and
00
belong to a class p if p.
0
/
D
p.
00
/
D
p,wherep
2
Œ0; 1. For each
two matchings
0
and
00
, such that p.
0
/<p.
00
/, we can compute their schema
matching level of certainty, ˝.
0
/ and ˝.
00
/. We say that a matching algorithm is
monotonic
if for any two such matchings p.
0
/<p.
00
/
!
˝.
0
/<˝.
00
/. Intu-
itively, a matching algorithm is monotonic if it ranks all possible schema matchings
according to their precision level.
A monotonic matching algorithm easily identifies the exact matching. Let
be
the exact matching, then p.
/
D
1. For any other matching
0
, p.
0
/<p.
/.
Therefore, if p.
0
/<p.
/, then from monotonicity ˝.
0
/<˝.
/. All one has
to do then is to devise a method for finding a matching
that maximizes ˝.
1
Figure
3.2
provides an illustration of the monotonicity principle using a matching
of a simplified version of two Web forms. Both schemata have nine attributes, all of
which are matched under the exact matching. Given a set of matchings, each value
Fig. 3.2
Illustration of the monotonicity principle
1
In
Gal et al.
[
2005a
], where the monotonicity principle was originally introduced, it was shown
that while such a method works well for fuzzy aggregators (e.g., weighted average) it does not
work for t-norms such as min.
