Graphics Reference
In-Depth Information
Φ
being
r
i
the rank of the observation
i
and
the cumulative normal function.
This transformation is useful to obtain a new variable that is very likely to behave
like to a normally distributed one. However, this transformation cannot be applied
separately to the training and test partitions [
30
]. Therefore, this transformation is
only recommended when the test and training data is the same.
3.5.6 Box-Cox Transformations
A big drawback when selecting the optimal transformation for an attribute is that
we do not know in advance which transformation will be the best to improve the
model performance. The Box-Cox transformation aims to transform a continuous
variable into an almost normal distribution. As [
30
] indicates, this can be achieved
by mapping the values using following the set of transformations:
x
λ
−
1
/λ,
λ
=
0
y
=
(3.29)
log
(
x
),
λ
=
0
All linear, inverse, quadratic and similar transformations are special cases of the
Box-Cox transformations. Please note that all the values of variable
x
in Eq. (
3.29
)
must be positive. If we have negative values in the attribute we must add a parameter
c
to offset such negative values:
(
)
λ
−
1
x
+
c
/
g
λ,
λ
=
0
y
=
(3.30)
log
(
x
+
c
)/
g
,
λ
=
0
The parameter
g
is used to scale the resulting values, and it is often considered as the
geometric mean of the data. The value of
λ
is iteratively found by testing different
values in the range from
0 in small steps until the resulting attribute is as
close as possible to the normal distribution.
In [
30
] a likelihood function to be maximized depending on the value of
−
3
.
0to3
.
is
defined based on the work of Johnson and Wichern [
19
]. This function is computed
as:
λ
⎡
⎣
⎤
⎦
+
(λ
−
m
m
n
2
ln
1
m
2
L
(λ)
=−
1
(
y
j
−
y
)
1
)
lnx
j
,
(3.31)
j
=
j
=
1
where
y
j
is the transformation of the value
x
j
using Eq. (
3.29
), and
y
is the mean of
the transformed values.
