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+% $Id: examples.Rnw 66 2010-09-03 08:50:26Z jranke $
+%%\VignetteIndexEntry{Examples for kinetic evaluations using mkin}
+%%VignetteDepends{FME}
+%%\usepackage{Sweave}
+\documentclass[12pt,a4paper]{article}
+\usepackage{a4wide}
+%%\usepackage[lists,heads]{endfloat}
+\input{header}
+\hypersetup{
+ pdftitle = {Examples for kinetic evaluations using mkin},
+ pdfsubject = {Manuscript},
+ pdfauthor = {Johannes Ranke},
+ colorlinks = {true},
+ linkcolor = {blue},
+ citecolor = {blue},
+ urlcolor = {red},
+ hyperindex = {true},
+ linktocpage = {true},
+}
+
+\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\\
+}
+\maketitle
+
+%\begin{abstract}
+%\end{abstract}
+
+
+\thispagestyle{empty} \setcounter{page}{0}
+
+\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
+
+\begin{Schunk}
+\begin{Sinput}
+R> library("mkin")
+R> 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))
+R> FOCUS_2006_L1_mkin <- mkin_wide_to_long(FOCUS_2006_L1)
+\end{Sinput}
+\end{Schunk}
+
+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}.
+
+\begin{Schunk}
+\begin{Sinput}
+R> SFO <- mkinmod(parent = list(type = "SFO"))
+R> FOMC <- mkinmod(parent = list(type = "FOMC"))
+R> DFOP <- mkinmod(parent = list(type = "DFOP"))
+\end{Sinput}
+\end{Schunk}
+
+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.
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L1.SFO <- mkinfit(SFO, FOCUS_2006_L1_mkin, quiet=TRUE)
+R> summary(m.L1.SFO)
+\end{Sinput}
+\begin{Soutput}
+mkin version: 0.9.10
+R version: 2.15.2
+Date of fit: Sat Feb 16 21:38:15 2013
+Date of summary: Sat Feb 16 21:38:15 2013
+
+Equations:
+[1] d_parent = - k_parent_sink * parent
+
+Starting values for optimised parameters:
+ initial type transformed
+parent_0 100.0 state 100.000000
+k_parent_sink 0.1 deparm -2.302585
+
+Fixed parameter values:
+None
+
+Optimised, transformed parameters:
+ Estimate Std. Error
+parent_0 92.471 1.368
+k_parent_sink -2.347 0.041
+
+Backtransformed parameters:
+ Estimate
+parent_0 92.471
+k_parent_sink 0.096
+
+Residual standard error: 2.948 on 16 degrees of freedom
+
+Chi2 error levels in percent:
+ err.min n.optim df
+All data 3.424 2 7
+parent 3.424 2 7
+
+Estimated disappearance times:
+ DT50 DT90
+parent 7.249 24.08
+
+Estimated formation fractions:
+ ff
+parent_sink 1
+
+Parameter correlation:
+ parent_0 k_parent_sink
+parent_0 1.0000 0.6248
+k_parent_sink 0.6248 1.0000
+
+Data:
+ time variable observed predicted residual
+ 0 parent 88.3 92.471 -4.1710
+ 0 parent 91.4 92.471 -1.0710
+ 1 parent 85.6 84.039 1.5610
+ 1 parent 84.5 84.039 0.4610
+ 2 parent 78.9 76.376 2.5241
+ 2 parent 77.6 76.376 1.2241
+ 3 parent 72.0 69.412 2.5884
+ 3 parent 71.9 69.412 2.4884
+ 5 parent 50.3 57.330 -7.0301
+ 5 parent 59.4 57.330 2.0699
+ 7 parent 47.0 47.352 -0.3515
+ 7 parent 45.1 47.352 -2.2515
+ 14 parent 27.7 24.247 3.4527
+ 14 parent 27.3 24.247 3.0527
+ 21 parent 10.0 12.416 -2.4163
+ 21 parent 10.4 12.416 -2.0163
+ 30 parent 2.9 5.251 -2.3513
+ 30 parent 4.0 5.251 -1.2513
+\end{Soutput}
+\end{Schunk}
+
+A plot of the fit is obtained with the plot function for mkinfit objects.
+
+\begin{Schunk}
+\begin{Sinput}
+R> plot(m.L1.SFO)
+\end{Sinput}
+\end{Schunk}
+\includegraphics{examples-L1_SFO_plot}
+
+The residual plot can be obtained using the information contained in the
+mkinfit object, which is in fact a derivative of an modFit object defined by
+the \Rpackage{FME} package.
+
+\begin{Schunk}
+\begin{Sinput}
+R> plot(m.L1.SFO$data$time, m.L1.SFO$data$residual,
++ xlab = "Time", ylab = "Residual", ylim = c(-8, 8))
+R> abline(h = 0, lty = 2)
+\end{Sinput}
+\end{Schunk}
+\includegraphics{examples-L1_SFO_residuals}
+
+For comparison, the FOMC model is fitted as well, and the $\chi^2$ error level
+is checked.
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L1.FOMC <- mkinfit(FOMC, FOCUS_2006_L1_mkin, quiet=TRUE)
+R> s.m.L1.FOMC <- summary(m.L1.FOMC)
+R> s.m.L1.FOMC$errmin
+\end{Sinput}
+\begin{Soutput}
+ err.min n.optim df
+All data 0.03618911 3 6
+parent 0.03618911 3 6
+\end{Soutput}
+\end{Schunk}
+
+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
+
+\begin{Schunk}
+\begin{Sinput}
+R> library("mkin")
+R> 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))
+R> FOCUS_2006_L2_mkin <- mkin_wide_to_long(FOCUS_2006_L2)
+\end{Sinput}
+\end{Schunk}
+
+Again, the SFO model is fitted and a summary is obtained.
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L2.SFO <- mkinfit(SFO, FOCUS_2006_L2_mkin, quiet=TRUE)
+R> summary(m.L2.SFO)
+\end{Sinput}
+\begin{Soutput}
+mkin version: 0.9.10
+R version: 2.15.2
+Date of fit: Sat Feb 16 21:38:15 2013
+Date of summary: Sat Feb 16 21:38:15 2013
+
+Equations:
+[1] d_parent = - k_parent_sink * parent
+
+Starting values for optimised parameters:
+ initial type transformed
+parent_0 100.0 state 100.000000
+k_parent_sink 0.1 deparm -2.302585
+
+Fixed parameter values:
+None
+
+Optimised, transformed parameters:
+ Estimate Std. Error
+parent_0 91.4656 3.807
+k_parent_sink -0.4112 0.107
+
+Backtransformed parameters:
+ Estimate
+parent_0 91.466
+k_parent_sink 0.663
+
+Residual standard error: 5.51 on 10 degrees of freedom
+
+Chi2 error levels in percent:
+ err.min n.optim df
+All data 14.38 2 4
+parent 14.38 2 4
+
+Estimated disappearance times:
+ DT50 DT90
+parent 1.046 3.474
+
+Estimated formation fractions:
+ ff
+parent_sink 1
+
+Parameter correlation:
+ parent_0 k_parent_sink
+parent_0 1.0000 0.4295
+k_parent_sink 0.4295 1.0000
+
+Data:
+ time variable observed predicted residual
+ 0 parent 96.1 91.4656079103 4.6344
+ 0 parent 91.8 91.4656079103 0.3344
+ 1 parent 41.4 47.1395280371 -5.7395
+ 1 parent 38.7 47.1395280371 -8.4395
+ 3 parent 19.3 12.5210295280 6.7790
+ 3 parent 22.3 12.5210295280 9.7790
+ 7 parent 4.6 0.8833842647 3.7166
+ 7 parent 4.6 0.8833842647 3.7166
+ 14 parent 2.6 0.0085318162 2.5915
+ 14 parent 1.2 0.0085318162 1.1915
+ 28 parent 0.3 0.0000007958 0.3000
+ 28 parent 0.6 0.0000007958 0.6000
+\end{Soutput}
+\end{Schunk}
+
+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.
+
+\begin{Schunk}
+\begin{Sinput}
+R> plot(m.L2.SFO)
+\end{Sinput}
+\end{Schunk}
+\includegraphics{examples-L2_SFO_plot}
+
+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.
+
+\begin{Schunk}
+\begin{Sinput}
+R> plot(m.L2.SFO$data$time, m.L2.SFO$data$residual,
++ xlab = "Time", ylab = "Residual", ylim = c(-10, 10))
+R> abline(h = 0, lty = 2)
+\end{Sinput}
+\end{Schunk}
+\includegraphics{examples-L2_SFO_residuals}
+
+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.
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L2.FOMC <- mkinfit(FOMC, FOCUS_2006_L2_mkin, quiet=TRUE)
+R> plot(m.L2.FOMC)
+R> s.m.L2.FOMC <- summary(m.L2.FOMC)
+R> s.m.L2.FOMC$errmin
+\end{Sinput}
+\begin{Soutput}
+ err.min n.optim df
+All data 0.06204245 3 3
+parent 0.06204245 3 3
+\end{Soutput}
+\end{Schunk}
+\includegraphics{examples-L2_FOMC}
+
+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.
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L2.DFOP <- mkinfit(DFOP, FOCUS_2006_L2_mkin, quiet=TRUE)
+R> plot(m.L2.DFOP)
+\end{Sinput}
+\end{Schunk}
+\includegraphics{examples-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.
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L2.DFOP <- mkinfit(DFOP, FOCUS_2006_L2_mkin,
++ parms.ini = c(k1 = 1, k2 = 0.01, g = 0.8),
++ quiet=TRUE)
+R> plot(m.L2.DFOP)
+R> summary(m.L2.DFOP)
+\end{Sinput}
+\begin{Soutput}
+mkin version: 0.9.10
+R version: 2.15.2
+Date of fit: Sat Feb 16 21:38:16 2013
+Date of summary: Sat Feb 16 21:38:16 2013
+
+Equations:
+[1] d_parent = - ((k1 * g * exp(-k1 * time) + k2 * (1 - g) * exp(-k2 * time)) / (g * exp(-k1 * time) + (1 - g) * exp(-k2 * time))) * parent
+
+Starting values for optimised parameters:
+ initial type transformed
+parent_0 1e+02 state 100.0000000
+k1 1e+00 deparm 0.0000000
+k2 1e-02 deparm -4.6051702
+g 8e-01 deparm 0.9802581
+
+Fixed parameter values:
+None
+
+Optimised, transformed parameters:
+ Estimate Std. Error
+parent_0 93.9500 NA
+k1 4.9589 NA
+k2 -1.0880 NA
+g -0.2821 NA
+
+Backtransformed parameters:
+ Estimate
+parent_0 93.950
+k1 142.434
+k2 0.337
+g 0.402
+
+Residual standard error: 1.732 on 8 degrees of freedom
+
+Chi2 error levels in percent:
+ err.min n.optim df
+All data 2.529 4 2
+parent 2.529 4 2
+
+Estimated disappearance times:
+ DT50 DT90
+parent NA NA
+
+Estimated formation fractions:
+[1] ff
+<0 rows> (or 0-length row.names)
+
+Data:
+ time variable observed predicted residual
+ 0 parent 96.1 93.950000 2.1500
+ 0 parent 91.8 93.950000 -2.1500
+ 1 parent 41.4 40.143423 1.2566
+ 1 parent 38.7 40.143423 -1.4434
+ 3 parent 19.3 20.464500 -1.1645
+ 3 parent 22.3 20.464500 1.8355
+ 7 parent 4.6 5.318322 -0.7183
+ 7 parent 4.6 5.318322 -0.7183
+ 14 parent 2.6 0.503070 2.0969
+ 14 parent 1.2 0.503070 0.6969
+ 28 parent 0.3 0.004501 0.2955
+ 28 parent 0.6 0.004501 0.5955
+\end{Soutput}
+\begin{Sinput}
+R> s.m.L2.DFOP <- summary(m.L2.DFOP)
+R> s.m.L2.DFOP$errmin
+\end{Sinput}
+\begin{Soutput}
+ err.min n.optim df
+All data 0.02528763 4 2
+parent 0.02528763 4 2
+\end{Soutput}
+\end{Schunk}
+\includegraphics{examples-L2_DFOP_2}
+
+Therefore, the FOMC model is clearly the best-fit model based on the
+$\chi^2$ error level criterion.
+
+\subsection{Laboratory Data L3}
+
+The following code defines example dataset L3 from the FOCUS kinetics
+report, p. 290
+
+\begin{Schunk}
+\begin{Sinput}
+R> library("mkin")
+R> 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))
+R> FOCUS_2006_L3_mkin <- mkin_wide_to_long(FOCUS_2006_L3)
+\end{Sinput}
+\end{Schunk}
+
+SFO model, summary and plot:
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L3.SFO <- mkinfit(SFO, FOCUS_2006_L3_mkin, quiet=TRUE)
+R> summary(m.L3.SFO)
+\end{Sinput}
+\begin{Soutput}
+mkin version: 0.9.10
+R version: 2.15.2
+Date of fit: Sat Feb 16 21:38:16 2013
+Date of summary: Sat Feb 16 21:38:16 2013
+
+Equations:
+[1] d_parent = - k_parent_sink * parent
+
+Starting values for optimised parameters:
+ initial type transformed
+parent_0 100.0 state 100.000000
+k_parent_sink 0.1 deparm -2.302585
+
+Fixed parameter values:
+None
+
+Optimised, transformed parameters:
+ Estimate Std. Error
+parent_0 74.873 8.458
+k_parent_sink -3.678 0.326
+
+Backtransformed parameters:
+ Estimate
+parent_0 74.873
+k_parent_sink 0.025
+
+Residual standard error: 12.91 on 6 degrees of freedom
+
+Chi2 error levels in percent:
+ err.min n.optim df
+All data 21.24 2 6
+parent 21.24 2 6
+
+Estimated disappearance times:
+ DT50 DT90
+parent 27.43 91.12
+
+Estimated formation fractions:
+ ff
+parent_sink 1
+
+Parameter correlation:
+ parent_0 k_parent_sink
+parent_0 1.0000 0.5484
+k_parent_sink 0.5484 1.0000
+
+Data:
+ time variable observed predicted residual
+ 0 parent 97.8 74.873 22.92734
+ 3 parent 60.0 69.407 -9.40654
+ 7 parent 51.0 62.734 -11.73403
+ 14 parent 43.0 52.563 -9.56336
+ 30 parent 35.0 35.083 -0.08281
+ 60 parent 22.0 16.439 5.56137
+ 91 parent 15.0 7.510 7.48961
+ 120 parent 12.0 3.609 8.39083
+\end{Soutput}
+\begin{Sinput}
+R> plot(m.L3.SFO)
+\end{Sinput}
+\end{Schunk}
+\includegraphics{examples-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:
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L3.FOMC <- mkinfit(FOMC, FOCUS_2006_L3_mkin, quiet=TRUE)
+R> plot(m.L3.FOMC)
+R> s.m.L3.FOMC <- summary(m.L3.FOMC)
+R> s.m.L3.FOMC$errmin
+\end{Sinput}
+\begin{Soutput}
+ err.min n.optim df
+All data 0.07321867 3 5
+parent 0.07321867 3 5
+\end{Soutput}
+\begin{Sinput}
+R> endpoints(m.L3.FOMC)
+\end{Sinput}
+\begin{Soutput}
+$distimes
+ DT50 DT90
+parent 7.729478 431.2428
+
+$ff
+logical(0)
+
+$SFORB
+logical(0)
+\end{Soutput}
+\end{Schunk}
+\includegraphics{examples-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:
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L3.DFOP <- mkinfit(DFOP, FOCUS_2006_L3_mkin, quiet=TRUE)
+R> plot(m.L3.DFOP)
+R> s.m.L3.DFOP <- summary(m.L3.DFOP)
+R> s.m.L3.DFOP$errmin
+\end{Sinput}
+\begin{Soutput}
+ err.min n.optim df
+All data 0.02223992 4 4
+parent 0.02223992 4 4
+\end{Soutput}
+\end{Schunk}
+\includegraphics{examples-L3_DFOP}
+
+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
+
+\begin{Schunk}
+\begin{Sinput}
+R> library("mkin")
+R> 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))
+R> FOCUS_2006_L4_mkin <- mkin_wide_to_long(FOCUS_2006_L4)
+\end{Sinput}
+\end{Schunk}
+
+SFO model, summary and plot:
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L4.SFO <- mkinfit(SFO, FOCUS_2006_L4_mkin, quiet=TRUE)
+R> summary(m.L4.SFO)
+\end{Sinput}
+\begin{Soutput}
+mkin version: 0.9.10
+R version: 2.15.2
+Date of fit: Sat Feb 16 21:38:17 2013
+Date of summary: Sat Feb 16 21:38:17 2013
+
+Equations:
+[1] d_parent = - k_parent_sink * parent
+
+Starting values for optimised parameters:
+ initial type transformed
+parent_0 100.0 state 100.000000
+k_parent_sink 0.1 deparm -2.302585
+
+Fixed parameter values:
+None
+
+Optimised, transformed parameters:
+ Estimate Std. Error
+parent_0 96.44 1.949
+k_parent_sink -5.03 0.080
+
+Backtransformed parameters:
+ Estimate
+parent_0 96.442
+k_parent_sink 0.007
+
+Residual standard error: 3.651 on 6 degrees of freedom
+
+Chi2 error levels in percent:
+ err.min n.optim df
+All data 3.288 2 6
+parent 3.288 2 6
+
+Estimated disappearance times:
+ DT50 DT90
+parent 106 352
+
+Estimated formation fractions:
+ ff
+parent_sink 1
+
+Parameter correlation:
+ parent_0 k_parent_sink
+parent_0 1.0000 0.5865
+k_parent_sink 0.5865 1.0000
+
+Data:
+ time variable observed predicted residual
+ 0 parent 96.6 96.44 0.1585
+ 3 parent 96.3 94.57 1.7324
+ 7 parent 94.3 92.13 2.1744
+ 14 parent 88.8 88.00 0.7972
+ 30 parent 74.9 79.26 -4.3589
+ 60 parent 59.9 65.14 -5.2376
+ 91 parent 53.5 53.18 0.3167
+ 120 parent 49.0 43.99 5.0054
+\end{Soutput}
+\begin{Sinput}
+R> plot(m.L4.SFO)
+\end{Sinput}
+\end{Schunk}
+\includegraphics{examples-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
+
+\begin{Schunk}
+\begin{Sinput}
+R> m.L4.FOMC <- mkinfit(FOMC, FOCUS_2006_L4_mkin, quiet=TRUE)
+R> plot(m.L4.FOMC)
+R> s.m.L4.FOMC <- summary(m.L4.FOMC)
+R> s.m.L4.FOMC$errmin
+\end{Sinput}
+\begin{Soutput}
+ err.min n.optim df
+All data 0.02027643 3 5
+parent 0.02027643 3 5
+\end{Soutput}
+\end{Schunk}
+\includegraphics{examples-L4_FOMC}
+
+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}
+% vim: set foldmethod=syntax:

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