The Box-Cox
transformation or power transformation is a general and widely used linearization
procedure when no theory exists, which indicates that a certain transformation
of the input and/or response variables will result in a more linear model [39].
The idea is to model a power of the response variable y as a linear function of x:
(27)
The
value of l, which fits the linear function
of x best, is estimated using the
available data of the pure analytes. After the estimation of l
and of the regression coefficients b
and b0, the response
variable can be transformed according to:
(28)
If l=0,
it is common to transform y
according to:
(29)
The Box-Cox
transformation (27) was determined for the measurements
of the single refrigerants of the calibration data set. For R22 l=0.68
and for R134a l=0.74 were estimated. Then the relative saturation
pressures of the refrigerants of the calibration and the validation data were
transformed according to expression (28) . Similar to section
6.1 PLS models were built for the transformed calibration data and then
the validation data were predicted whereby the optimal number of principal components
was determined by the minimum error of crossvalidation of the calibration data.
The optimal model for R22 contained 11 principal components and the model for
R134a used 10 principal components. The calibration data were predicted with
a relative RMSE of 2.97% for R22 and 4.50% for R134a. The prediction of the
validation data, which is also shown in figure 34
was performed with rel. RMSE of 3.09% for R22 and 5.04% for R134a. Both, the
Durbin-Watson Statistics and the Wald-Wolfowitz Runs test are significant at
the 5% error level. In figure 34, it is visible
that the prediction of both analytes shows slightly a wave. Compared with the
standard PLS the Box-Cox Transformation allows a highly improved calibration
while a few nonlinearities remain uncalibrated. Although a rather high number
of principal components are needed, only a slight overfitting can be observed,
as the errors of the validation data are only moderately higher than the errors
of the calibration data.
figure 34: True-predicted plots
of the PLS for the validation data. The data were linearized by a Box-Cox transformation.
Ph. D. Thesis
