the residuals of the predictions of both analytes by all PLS models (Martens'
Uncertainty Test and minimum crossvalidation criterion), the Wald-Wolfowitz
Runs test (p=0.000) shows that
the sequence of the residuals is highly non-random. The null hypotheses of the
Durbin-Watson Statistics have to be rejected at the 5% error level for both
analytes and both models indicating a correlation of the residuals. The
combination of both statistics and the true-predicted plots indicate that the
PLS cannot deal with nonlinearities, which are present in the relationship
between the concentrations of the analytes and the signals of the device.
which are somehow visible in figure 22,
can be made more prominent by plotting the first and second partial derivatives
of the signals with respect to the relative saturation pressure versus the relative
saturation pressure of the analyte R22 and versus time in figure
33. It is obvious that during exposure to R22 the first partial derivative
is not constant along the concentration axis (y-axis)
confirming that the relationship between the sensor signals and the relative
saturation pressures of R22 is not linear. The partial derivative of the second
order is neither zero nor constant. This means that the relationship also cannot
be exactly described by polynomials with quadratic terms. The same nonlinear
relationship between the signal and the relative saturation pressure can also
be observed for R134a, but is not shown here. The nonlinearity can be ascribed
to a saturation effect, as the number of micropores is limited (see discussion
in section 3.2). For the high-concentrated mixtures,
the nonlinearity is even worse since the total concentrations add up to 0.2
pi/pio. As for both analytes this nonlinearity can be
observed and as a model for the nonlinear relationship between the concentrations
and the signals is not known, several extensions to the linear models and several
variable transformations are used for the calibration of this data set in the
next sections. Additionally, "true nonlinear" methods, the neural
networks, are applied to the data set in the last sections of this chapter.
figure 33: First and second partial
derivatives of the LOESS plot of figure 22
for R22 with respect to the relative saturation pressure.