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  pdftitle = {Examples for kinetic evaluations using kinfit},
  pdfsubject = {Manuscript},
  pdfauthor = {Johannes Ranke},
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\begin{document}
\title{Examples for kinetic evaluations using kinfit}
\author{\textbf{Johannes Ranke} \\[0.5cm]
%EndAName
Eurofins Regulatory AG\\
Weidenweg 15, CH--4310 Rheinfelden, Switzerland\\[0.5cm]
and\\[0.5cm]
University of Bremen\\
}
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%\begin{abstract}
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\tableofcontents

\textbf{Key words}: Kinetics, FOCUS, nonlinear optimisation

\section{Kinetic evaluations for parent compounds}
\label{intro}

These examples are also evaluated in a parallel vignette of the
\Rpackage{mkin} package \citep{pkg:mkin}. The datasets are from Appendix 3,
of the FOCUS kinetics report \citep{FOCUS2006, FOCUSkinetics2011}.

\subsection{Laboratory Data L1}

The following code defines an object containing the example dataset L1 from the
FOCUS kinetics report, p. 284

<<FOCUS_2006_L1_data, echo=TRUE, eval=TRUE>>=
library("kinfit")
FOCUS_2006_L1 = kinobject("Parent", "Degradation data", "")
FOCUS_2006_L1$data = data.frame(
  t = rep(c(0, 1, 2, 3, 5, 7, 14, 21, 30), each = 2),
  parent = c(88.3, 91.4, 85.6, 84.5, 78.9, 77.6, 
             72.0, 71.9, 50.3, 59.4, 47.0, 45.1,
             27.7, 27.3, 10.0, 10.4, 2.9, 4.0))
@

The following two lines fit the model and produce the summary report
of the model fit. This covers the numerical analyses given in the 
FOCUS report.

<<L1, echo=TRUE>>=
FOCUS_2006_L1$fits <- kinfit(FOCUS_2006_L1$data, 
  kinmodels = c("SFO", "FOMC", "DFOP"))
FOCUS_2006_L1$results <- kinresults(FOCUS_2006_L1$fits)
kinreport(FOCUS_2006_L1)
@

Obviously, the FOMC model and the DFOP model were not fitted. As discussed in the
kinfit vignette of this package, this occurs when the SFO model fits very well.

We can try to force the FOMC fit using the parameters obtained using mkin.

<<L1_2, echo=TRUE>>=
FOCUS_2006_L1$fits <- kinfit(FOCUS_2006_L1$data, 
  kinmodels = c("SFO", "FOMC", "DFOP"),
  start.FOMC = list(parent.0 = 92.47, alpha = 1.35e11, beta = 1.41e12))
FOCUS_2006_L1$results <- kinresults(FOCUS_2006_L1$fits)
kinreport(FOCUS_2006_L1)
@

It still does not converge. As discussed in the kinfit vignette, the FOMC model usually
is not returned by kinfit when the SFO model fits very well. This should be seen as 
a feature, not a bug, as the FOMC model is ill-defined in such cases.

A plot of the fit is obtained with the kinplot function.

<<L1_SFO_plot, fig=TRUE, echo=TRUE>>=
kinplot(FOCUS_2006_L1, ylab = "Observed")
@

The residual plot can be easily obtained by

<<L1_SFO_residuals, fig=TRUE, echo=TRUE>>=
kinresplot(FOCUS_2006_L1, "SFO", ylab = "Observed")
@

\subsection{Laboratory Data L2}

The following code defines example dataset L2 from the FOCUS kinetics
report, p. 287

<<FOCUS_2006_L2_data, echo=TRUE, eval=TRUE>>=
FOCUS_2006_L2 = kinobject("Parent", "Degradation data", "")
FOCUS_2006_L2$data = data.frame(
  t = rep(c(0, 1, 3, 7, 14, 28), each = 2),
  parent = c(96.1, 91.8, 41.4, 38.7,
             19.3, 22.3, 4.6, 4.6,
             2.6, 1.2, 0.3, 0.6))
@

Again, the SFO, FOMC and DFOP models are fitted and a report is printed.

<<L2, echo=TRUE>>=
FOCUS_2006_L2$fits <- kinfit(FOCUS_2006_L2$data, 
  kinmodels = c("SFO", "FOMC", "DFOP"))
FOCUS_2006_L2$results <- kinresults(FOCUS_2006_L2$fits)
kinreport(FOCUS_2006_L2)
@

Here, only the DFOP did not converge using default parameters. The DFOP fit can be 
obtained using refined starting parameters:

<<L2_2, echo=TRUE>>=
FOCUS_2006_L2$fits <- kinfit(FOCUS_2006_L2$data, 
  kinmodels = c("SFO", "FOMC", "DFOP"),
  start.DFOP = list(parent.0 = 94, g = 0.4, k1 = 142, k2 = 0.34))
FOCUS_2006_L2$results <- kinresults(FOCUS_2006_L2$fits)
kinreport(FOCUS_2006_L2)
@

Again, even with starting parameters very close to the optimum obtained using mkin, 
there is no convergence with kinfit. However, when looking at the fit obtained using
mkin plotted in the mkin vignette, it is clear that the point where the break point 
of the curve, caused by the large difference between k1 and k2, is not clearly defined
by the data. Therefore, it should be seen as a desirable feature of the
underlying nls() function that no solution is returned.

Comparison of $\chi^2$ error levels of the two models shows that the FOMC model allows
for a better representation of the data.  This is also obvious from the plot
of the fits.

<<L2_plot, fig=TRUE, echo=TRUE>>=
kinplot(FOCUS_2006_L2, ylab = "Observed")
@

Residual plots are obtained using kinresplot.

<<L2_resplot, fig=TRUE, echo=TRUE>>=
par(mfrow=c(2,1))
kinresplot(FOCUS_2006_L2, "SFO", ylab = "Observed")
kinresplot(FOCUS_2006_L2, "FOMC", ylab = "Observed")
@

\subsection{Laboratory Data L3}

The following code defines example dataset L3 from the FOCUS kinetics
report, p. 290 and attempts to fit the SFO, FOMC and DFOP models.

<<FOCUS_2006_L3, echo=TRUE, eval=TRUE>>=
FOCUS_2006_L3 = kinobject("Parent", "Degradation data", "")
FOCUS_2006_L3$data = data.frame(
  t = c(0, 3, 7, 14, 30, 60, 91, 120),
  parent = c(97.8, 60, 51, 43, 35, 22, 15, 12))
FOCUS_2006_L3$fits <- kinfit(FOCUS_2006_L3$data, 
  kinmodels = c("SFO", "FOMC", "DFOP"))
FOCUS_2006_L3$results <- kinresults(FOCUS_2006_L3$fits)
kinreport(FOCUS_2006_L3)
@

In this case, the FOMC model does not return a solution using kinfit. Trying with 
closer starting parameters gives success this time.

<<FOCUS_2006_L3_2, echo=TRUE, eval=TRUE, fig=TRUE>>=
FOCUS_2006_L3$fits <- kinfit(FOCUS_2006_L3$data, 
  kinmodels = c("SFO", "FOMC", "DFOP"),
  start.FOMC = list(parent.0 = 100, alpha = 0.5, beta = 2))
FOCUS_2006_L3$results <- kinresults(FOCUS_2006_L3$fits)
kinreport(FOCUS_2006_L3)
kinplot(FOCUS_2006_L3, ylab = "Observed")
@

Based on the $\chi^2$ error level criterion and the visual analysis of the
fits, the DFOP model would be the best-fit model of choice for laboratory data
L3.

\subsection{Laboratory Data L4}

The following code defines example dataset L4 from the FOCUS kinetics
report, p. 293 and attempts to fit the SFO, FOMC and DFOP models.

<<FOCUS_2006_L4, echo=TRUE, eval=TRUE, fig=TRUE>>=
FOCUS_2006_L4 = kinobject("Parent", "Degradation data", "")
FOCUS_2006_L4$data = data.frame(
  t = c(0, 3, 7, 14, 30, 60, 91, 120),
  parent = c(96.6, 96.3, 94.3, 88.8, 74.9, 59.9, 53.5, 49.0))
FOCUS_2006_L4$fits <- kinfit(FOCUS_2006_L4$data, 
  kinmodels = c("SFO", "FOMC", "DFOP"))
FOCUS_2006_L4$results <- kinresults(FOCUS_2006_L4$fits)
kinreport(FOCUS_2006_L4)
kinplot(FOCUS_2006_L4, ylab = "Observed")
@

Although the $\chi^2$ error level is slightly smaller for the DFOP model and also
for the FOMC model, the differences are small, and the SFO model may appear to
be a suitable choice. The better fit of the DFOP model depends very much on the
last three data points.

\bibliographystyle{plainnat}
\bibliography{references}

\end{document}
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