Graphics Reference
In-Depth Information
3.5.1 Linear Transformations
In the area of scientific discoveries and machine control, normalizations may not be
enough to adapt the data to improve the generated model. In these cases aggregating
the information contained in various attributesmight be beneficial. A family of simple
methods that can be used to this purpose are linear transformations. They are based
on simple algebraic transformations such as sums, averages, rotations, translations
and so on. Let
A
B
m
be a subset of the complete set of attributes
A
. If the following expression is applied
=
A
1
,
A
2
,...,
A
n
be a set of attributes, let
B
=
B
1
,
B
2
,...,
Z
=
r
1
B
1
+
r
2
B
2
+···+
r
m
B
M
(3.16)
a new derived attribute is constructed by taking a linear combination of attributes in
B
.
A special case arises when the
m
values are set to
r
1
=
r
2
= ··· =
r
m
=
1
/
m
,
that situation averages the considered attributes in
B
:
Z
=
(
B
1
+
B
2
+···+
B
m
)/
m
.
(3.17)
3.5.2 Quadratic Transformations
In quadratic transformations a new attribute is built as follows:
r
1
,
1
B
1
+
r
m
,
m
B
m
,
Z
=
r
1
,
2
B
1
B
2
+···+
r
m
−
1
,
m
B
m
−
1
B
m
+
(3.18)
where
r
i
,
j
is a real number. These kinds of transformations have been thoroughly
studied and can help to transform data to make it separable.
In Table
3.1
we show an example of how quadratic transformations can help us
to reveal knowledge that it is not explicitly present using the initial attributes of the
data set. Let us consider a set of conic sections described by the coefficients of the
algebraic expression that follows:
A
1
x
2
A
3
y
2
+
A
2
xy
+
+
A
4
x
+
A
5
y
+
A
6
=
0
.
(3.19)
From Eq. (
3.19
) there is no obvious interpretation that can be used to label the tuples
to any type of conic section. However, computing the quadratic transformation known
as the discriminant given by
A
2
−
Z
=
(
4
A
1
A
3
),
(3.20)
the sign of
Z
provides enough information to correctly label the tuples. Without the
new derived attribute
Z
we could not be able to classify them.
