models were built for the calibration data set of the refrigerants (see section
126.96.36.199) with Martens' Uncertainty Test ,
for the determination of the insignificant variables and with the recommended
determination of the number of principal components (factors) according to:
with a as the current dimensionality (principal
component number), Vytot as the
total residual y-variance validated
with a principal components
respectively 0 principal components (total y-variance).
This means that for each principal component 1% of the total y-variance is added to the variance at
principal component number a.
This ensures that only principal components are added, which significantly
improve the model.
a model was calibrated with 2 principal components and for R134a a model with
3 principal components. Martens' Uncertainty Test removed no variable for R22
and only the time points 15 s and 67 s for R134a. In figure
28, the x and y
loadings are shown for R134a. It is visible that nearly all variables are highly
correlated whereby the time points during desorption form a cluster (top right)
and the time points during sorption form a second cluster (bottom left). The
three dots near the origin of coordinates are the first 3 time points during
sorption and the remaining 2 dots are the fist 2 time-points during desorption.
The corresponding loadings plot for R22 is analogous with only the location
of the two clusters exchanged.
figure 28: Loadings plot of the
x and y variables for the calibration of R134a.
root mean square errors of the predictions of the calibration data are very
high with 11.89% for R22 and 11.40% for R134a. The prediction of the independent
validation set also showed disappointing high relative errors of 10.27% for
R22 and 9.94% for R134a. In figure 29 the predictions
of the validation data are shown as true-predicted plots. It is visible that
the plots show a curvature and consequently the predictions are highly biased.
The curved true-predicted plots are a typical sign of nonlinearities in the
data, which are not successfully calibrated by a linear model. In figure
30, the T scores are plotted versus the U scores of the model of R134a for
the first principal component, which explains 92.5% of the y-variance
and 54.2% of the x-variance. The different concentration levels
of R134a are visible along the axis of the U scores and the different concentrations
of the interfering analyte R22 are visible as scattering along the axis of the
T scores. As this type of plot shows the inner relationship of the PLS model
(see expression (10)), the
nonlinearities, which are visible in figure 30 and
which are also visible in the less scattered T scores versus U scores plots
for the higher principal components, demonstrate that the linear PLS model cannot
deal with a nonlinear relationship present in the data. The corresponding plots
for R22 are analogous and will not be discussed here.
figure 29: True-predicted plots
of the PLS calibration for the validation data. The optimal number of principal
components was determined by Martens' Uncertainty. The predictions are poor
due to systematic biases.
figure 30: T scores versus U scores
for the first principal component of the model for the calibration of R134a.
It is known , - that the linear PLS sometimes can be successfully
applied to nonlinear problems when minor important higher principal components
are included into the model. These principal components may encapsulate not
only noise but also significant information about the nonlinear relationship.
Thus, the minimum crossvalidation error was used as selection criterion for
the optimal number of principal components, as this criterion is less conservative
in terms of the number of principal components compared to expression
(25). The optimal models contain 5 components for R22 and 6 components for R134a,
which were selected using the minima in figure 31.
The calibration data were predicted with relative RMSE of 10.47% for R22 and
8.51% for R134a. The predictions of the validation data are visualized in figure
32 with relative RMSE of 8.69% for R22 and 7.63% for R134a. Although the
increasing number of principal components improved the errors of prediction,
the plots show that the improvement can be put down to less scattered predictions
but not to an improved calibration of the nonlinearities. The true-predicted
plots can also explain why the predictions of the calibration data are worse
than the predictions of the validation data. At both ends of the concentration
range, the predictions are most biased. As the concentration range of the calibration
data is wider (pi/pio=0-0.1) than the concentration range
of the validation data (pi/pio=0.05-0.095) the high bias
at the lower and upper concentration limit significantly increases the RMSE
of the prediction of the complete calibration data set.
figure 31: Crossvalidated root
mean square errors for the first 20 principal components for the calibration
of R22 and R134a.
True-predicted plots of the PLS for the validation data. The optimal number
of principal components was determined by a full crossvalidation of the calibration
data. The predictions are still rather poor visible as systematic biases.
to find a statistical basis to evaluate the ability of a method to calibrate
nonlinearities, statistical methods can be used, which investigate the randomness
of residuals of sequential observations. Theses tests like the Durbin-Watson
statistics were originally developed for sequential observations equidistant
in space or time. Centner et al.  showed that the Durbin-Watson
Statistics can be used to test the residuals of purity data without a space
or time component while implementing the test into a SIMPLISIMA algorithm for
a mixture analysis. In this study, the mean residuals in figure
29 and figure 32 are treated with these statistics
by using the increasing concentration levels as "virtual equidistant space
component". The Durbin-Watson statistics is used to investigate the correlation
of the residuals and additionally the Wald-Wolfowitz Runs test is used to test
if the signs of the residuals occur in a random sequence.