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  pdftitle = {Examples for kinetic evaluations using mkin},
  pdfsubject = {Manuscript},
  pdfauthor = {Johannes Ranke},
  colorlinks = {true},
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\begin{document}
\title{Examples for kinetic evaluations using mkin}
\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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\clearpage

\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{kinfit} package \citep{pkg:kinfit}. The datasets are from Appendix 3,
of the FOCUS kinetics report \citep{FOCUS2006, FOCUSkinetics2011}.

\subsection{Laboratory Data L1}

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

<<FOCUS_2006_L1_data, echo=TRUE, eval=TRUE>>=
library("mkin")
FOCUS_2006_L1 = 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))
FOCUS_2006_L1_mkin <- mkin_wide_to_long(FOCUS_2006_L1)
@

The next step is to set up the models used for the kinetic analysis. Note that
the model definitions contain the names of the observed variables in the data.
In this case, there is only one variable called \Robject{parent}.

<<Simple_models, echo=TRUE>>=
SFO <- mkinmod(parent = list(type = "SFO"))
FOMC <- mkinmod(parent = list(type = "FOMC"))
DFOP <- mkinmod(parent = list(type = "DFOP"))
@

The three models cover the first assumption of simple first order (SFO),
the case of declining rate constant over time (FOMC) and the case of two
different phases of the kinetics (DFOP). For a more detailed discussion
of the models, please see the FOCUS kinetics report.

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

<<L1_SFO, echo=TRUE>>=
m.L1.SFO <- mkinfit(SFO, FOCUS_2006_L1_mkin, quiet=TRUE)
summary(m.L1.SFO)
@

A plot of the fit is obtained with the plot function for mkinfit objects.

<<L1_SFO_plot, fig=TRUE, echo=TRUE>>=
plot(m.L1.SFO)
@

The residual plot can be easily obtained by

<<L1_SFO_residuals, fig=TRUE, echo=TRUE>>=
mkinresplot(m.L1.SFO, ylab = "Observed", xlab = "Time")
@

For comparison, the FOMC model is fitted as well, and the $\chi^2$ error level
is checked.

<<L1_FOMC, echo=TRUE>>=
m.L1.FOMC <- mkinfit(FOMC, FOCUS_2006_L1_mkin, quiet=TRUE)
s.m.L1.FOMC <- summary(m.L1.FOMC)
s.m.L1.FOMC$errmin
@

Due to the higher number of parameters, and the lower number of degrees of freedom
of the fit, the $\chi^2$ error level is actually higher for the FOMC model (3.6\%) than 
for the SFO model (3.4\%).

\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 = 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))
FOCUS_2006_L2_mkin <- mkin_wide_to_long(FOCUS_2006_L2)
@

Again, the SFO model is fitted and a summary is obtained.

<<L2_SFO, echo=TRUE>>=
m.L2.SFO <- mkinfit(SFO, FOCUS_2006_L2_mkin, quiet=TRUE)
summary(m.L2.SFO)
@

The $\chi^2$ error level of 14\% suggests that the model does not fit very well.
This is also obvious from the plots of the fit and the residuals.

<<L2_SFO_plot, fig=TRUE, echo=TRUE>>=
plot(m.L2.SFO)
@

In the FOCUS kinetics report, it is stated that there is no apparent systematic
error observed from the residual plot up to the measured DT90 (approximately at
day 5), and there is an underestimation beyond that point.

<<L2_SFO_residuals, fig=TRUE, echo=TRUE>>=
mkinresplot(m.L2.SFO, ylab = "Observed", xlab = "Time")
@

We may add that it is difficult to judge the random nature of the residuals just 
from the three samplings at days 0, 1 and 3. Also, it is not clear why a
consistent underestimation after the approximate DT90 should be irrelevant.

For comparison, the FOMC model is fitted as well, and the $\chi^2$ error level
is checked.

<<L2_FOMC, echo=TRUE, fig=TRUE>>=
m.L2.FOMC <- mkinfit(FOMC, FOCUS_2006_L2_mkin, quiet=TRUE)
plot(m.L2.FOMC)
s.m.L2.FOMC <- summary(m.L2.FOMC)
s.m.L2.FOMC$errmin
@

The error level at which the $\chi^2$ test passes is much lower in this case.
Therefore, the FOMC model provides a better description of the data, as less
experimental error has to be assumed in order to explain the data.

Fitting the four parameter DFOP model does not further reduce the 
$\chi^2$ error level. 

<<L2_DFOP, echo=TRUE, fig=TRUE>>=
m.L2.DFOP <- mkinfit(DFOP, FOCUS_2006_L2_mkin, quiet=TRUE)
plot(m.L2.DFOP)
@

Here, the default starting parameters for the DFOP model obviously do not lead
to a reasonable solution. Therefore the fit is repeated with different starting
parameters.

<<L2_DFOP_2, echo=TRUE, fig=TRUE>>=
m.L2.DFOP <- mkinfit(DFOP, FOCUS_2006_L2_mkin, 
  parms.ini = c(k1 = 1, k2 = 0.01, g = 0.8),
  quiet=TRUE)
plot(m.L2.DFOP)
summary(m.L2.DFOP)
s.m.L2.DFOP <- summary(m.L2.DFOP)
s.m.L2.DFOP$errmin
@

Therefore, the FOMC model is clearly the best-fit model based on the 
$\chi^2$ error level criterion.

\subsection{Laboratory Data L3}

The following code defines example dataset L3 from the FOCUS kinetics
report, p. 290

<<FOCUS_2006_L3_data, echo=TRUE, eval=TRUE>>=
FOCUS_2006_L3 = 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_mkin <- mkin_wide_to_long(FOCUS_2006_L3)
@

SFO model, summary and plot:

<<L3_SFO, echo=TRUE, fig=TRUE>>=
m.L3.SFO <- mkinfit(SFO, FOCUS_2006_L3_mkin, quiet=TRUE)
summary(m.L3.SFO)
plot(m.L3.SFO)
@

The $\chi^2$ error level of 22\% as well as the plot suggest that the model
does not fit very well. 

The FOMC model performs better:

<<L3_FOMC, echo=TRUE, fig=TRUE>>=
m.L3.FOMC <- mkinfit(FOMC, FOCUS_2006_L3_mkin, quiet=TRUE)
plot(m.L3.FOMC)
s.m.L3.FOMC <- summary(m.L3.FOMC)
s.m.L3.FOMC$errmin
endpoints(m.L3.FOMC)
@

The error level at which the $\chi^2$ test passes is 7\% in this case.

Fitting the four parameter DFOP model further reduces the $\chi^2$ error level
considerably:

<<L3_DFOP, echo=TRUE, fig=TRUE>>=
m.L3.DFOP <- mkinfit(DFOP, FOCUS_2006_L3_mkin, quiet=TRUE)
plot(m.L3.DFOP)
s.m.L3.DFOP <- summary(m.L3.DFOP)
s.m.L3.DFOP$errmin
@

Therefore, the DFOP model is the best-fit model based on the $\chi^2$ error
level criterion for laboratory data L3.

\subsection{Laboratory Data L4}

The following code defines example dataset L4 from the FOCUS kinetics
report, p. 293

<<FOCUS_2006_L4_data, echo=TRUE, eval=TRUE>>=
FOCUS_2006_L4 = 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_mkin <- mkin_wide_to_long(FOCUS_2006_L4)
@

SFO model, summary and plot:

<<L4_SFO, echo=TRUE, fig=TRUE>>=
m.L4.SFO <- mkinfit(SFO, FOCUS_2006_L4_mkin, quiet=TRUE)
summary(m.L4.SFO)
plot(m.L4.SFO)
@

The $\chi^2$ error level of 3.3\% as well as the plot suggest that the model
fits very well. 

The FOMC model for comparison

<<L4_FOMC, echo=TRUE, fig=TRUE>>=
m.L4.FOMC <- mkinfit(FOMC, FOCUS_2006_L4_mkin, quiet=TRUE)
plot(m.L4.FOMC)
s.m.L4.FOMC <- summary(m.L4.FOMC)
s.m.L4.FOMC$errmin
@

The error level at which the $\chi^2$ test passes is slightly lower for the FOMC 
model. However, the difference appears negligible.

\bibliographystyle{plainnat}
\bibliography{references}

\end{document}
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