Hardware Reference
In-Depth Information
4.3.3
Pre-Processing and Scaling Data
Data used to derive analytical models, also if originated from the same source/mod-
eled architecture, due to different positions in the design space can have different
values distribution and orders of magnitudes. Especially when the case is the latter,
to create better prediction it is better to pre-process and/or scale data.
Data transformation is very important because in most of the cases the analytical
models used to predict the data work better when data distribution follows some rules.
As an example, if we consider an analytical model that uses the standard deviation
of the training data to predict unknown data, this standard deviation values can be
very high if the data distribution is skewed. In this case, it is highly recommended to
first transform the data to approach a better symmetry and then to perform the model
training and related prediction.
Box-Cox power transformation. A powerful transformation adopted in the
above-mentioned cases is called Box-Cox power transformation [ 4 ]. The Box-Cox
power transformation is a useful data pre-processing technique used to reduce data
variation, make the data more normal distribution-like and improve the correlation
between variables. The power transformation is defined as a continuously varying
function, with respect to the power parameter λ :
( y k
1) , f λ
=
0
y ( λ )
k
=
(4.7)
log y k ,
if λ
=
0
In the validation results of the models that we adopted in this topic, we considered a
family of transformations as potential candidates λ {
. All the Box-Cox
power transformations are only defined with positive values. In case of negative
values, a constant value has to be added in order to make them positive. To keep the
prediction consistent with the actual objective functions of the target problem, an
inverse Box-Cox transformation has been applied on the predicted data.
Some care has to be taken when performing the inverse Box-Cox transformation,
since the condition λy ( λ )
k
1, 0 . 5, 0,
1
}
0) has to be satisfied. Therefore
possible unfeasible outcomes has to be taken into account.
Centering and scaling data. Another pre-processing step that is usually applied
to the data after the data transformation is the centering and scaling step. The goal
of this step is to remove the bias from the input data (mean equal to zero) and to
standardize the variance (standard deviation equal to 1). This transformation is also
called “Autoscaling”. When the autoscaling transformation is applied to a set of data,
from each value the mean value is removed and it is scaled by the standard deviation:
y autoscaled
+
1 > 0 (when λ
=
μ y ) y . Another common alternative step is to normalize
data in the unitary interval [0, 1]: y normalized
=
( y original
min y ).
Usually this normalization step is performed on input variables too: in this way data
result to be well scaled, being perfectly comparable as regards range extensions. This
solution prevent numeric issues that usually arise during RSM training in presence
of different scaled variables.
=
( y original
min y ) / (max y
Search WWH ::




Custom Search