Further formatting improvement of benchmark vignette

Also, use .rmd extension instead of .Rmd for vignettes.

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+---
+title: Introduction to mkin
+author: Johannes Ranke
+date: "`r Sys.Date()`"
+output:
+  html_document:
+    toc: true
+    toc_float: true
+    code_folding: hide
+    fig_retina: null
+bibliography: references.bib
+vignette: >
+  %\VignetteEngine{knitr::rmarkdown}
+  %\VignetteIndexEntry{mkin - Kinetic evaluation of chemical degradation data}
+  %\VignetteEncoding{UTF-8}
+---
+
+[Wissenschaftlicher Berater, Kronacher Str. 12, 79639 Grenzach-Wyhlen, Germany](http://www.jrwb.de)<br />
+[Privatdozent at the University of Bremen](http://chem.uft.uni-bremen.de/ranke)
+
+```{r, include = FALSE}
+require(knitr)
+opts_chunk$set(engine='R', tidy=FALSE)
+```
+
+# Abstract
+In the regulatory evaluation of chemical substances like plant protection
+products (pesticides), biocides and other chemicals, degradation data play an
+important role. For the evaluation of pesticide degradation experiments,
+detailed guidance has been developed, based on nonlinear optimisation.
+The `R` add-on package `mkin` implements fitting some of the models
+recommended in this guidance from within R and calculates some statistical
+measures for data series within one or more compartments, for parent and
+metabolites.
+
+```{r, echo = TRUE, cache = TRUE, fig = TRUE, fig.width = 8, fig.height = 7}
+library("mkin", quietly = TRUE)
+# Define the kinetic model
+m_SFO_SFO_SFO <- mkinmod(parent = mkinsub("SFO", "M1"),
+                         M1 = mkinsub("SFO", "M2"),
+                         M2 = mkinsub("SFO"),
+                         use_of_ff = "max", quiet = TRUE)
+
+
+# Produce model predictions using some arbitrary parameters
+sampling_times = c(0, 1, 3, 7, 14, 28, 60, 90, 120)
+d_SFO_SFO_SFO <- mkinpredict(m_SFO_SFO_SFO,
+  c(k_parent = 0.03,
+    f_parent_to_M1 = 0.5, k_M1 = log(2)/100,
+    f_M1_to_M2 = 0.9, k_M2 = log(2)/50),
+  c(parent = 100, M1 = 0, M2 = 0),
+  sampling_times)
+
+# Generate a dataset by adding normally distributed errors with
+# standard deviation 3, for two replicates at each sampling time
+d_SFO_SFO_SFO_err <- add_err(d_SFO_SFO_SFO, reps = 2,
+                             sdfunc = function(x) 3,
+                             n = 1, seed = 123456789 )
+
+# Fit the model to the dataset
+f_SFO_SFO_SFO <- mkinfit(m_SFO_SFO_SFO, d_SFO_SFO_SFO_err[[1]], quiet = TRUE)
+
+# Plot the results separately for parent and metabolites
+plot_sep(f_SFO_SFO_SFO, lpos = c("topright", "bottomright", "bottomright"))
+```
+
+# Background
+
+Many approaches are possible regarding the evaluation of chemical degradation
+data.
+
+The `mkin` package [@pkg:mkin] implements the approach recommended in the
+kinetics report provided by the FOrum for Co-ordination of pesticide fate
+models and their USe [@FOCUS2006; -@FOCUSkinetics2014]
+for simple decline data series, data series with transformation products,
+commonly termed metabolites, and for data series for more than one compartment.
+It is also possible to include back reactions, so equilibrium reactions and
+equilibrium partitioning can be specified, although this oftentimes leads to an
+overparameterisation of the model.
+
+When the first `mkin` code was published in 2010, the most commonly used tools for
+fitting more complex kinetic degradation models to experimental data were
+KinGUI [@schaefer2007], a MATLAB based tool with a graphical user
+interface that was specifically tailored to the task and included some output
+as proposed by the FOCUS Kinetics Workgroup, and ModelMaker, a general purpose
+compartment based tool providing infrastructure for fitting dynamic simulation
+models based on differential equations to data.
+
+The code was first uploaded to the BerliOS platform. When this was taken down,
+the version control history was imported into the R-Forge site (see *e.g.*
+[the initial commit on 11 May 2010](http://cgit.jrwb.de/mkin/commit/?id=30cbb4092f6d2d3beff5800603374a0d009ad770)),
+where the code is still occasionally updated.
+
+At that time, the R package `FME` (Flexible Modelling Environment)
+[@soetaert2010] was already available, and provided a good basis for
+developing a package specifically tailored to the task. The remaining challenge
+was to make it as easy as possible for the users (including the author of this
+vignette) to specify the system of differential equations and to include the
+output requested by the FOCUS guidance, such as the relative standard deviation
+that has to be assumed for the residuals, such that the $\chi^2$
+goodness-of-fit test as defined by the FOCUS kinetics workgroup would pass
+using an significance level $\alpha$ of 0.05. This relative error, expressed
+as a percentage, is often termed $\chi^2$ error level or similar.
+
+Also, `mkin` introduced using analytical solutions for parent only kinetics for
+improved optimization speed. Later, Eigenvalue based solutions were
+introduced to `mkin` for the case of linear differential equations (*i.e.*
+where the FOMC or DFOP models were not used for the parent compound), greatly
+improving the optimization speed for these cases. This, however, has become
+somehow obsolete, as the use of compiled code described below gives even
+smaller execution times.
+
+The possibility to specify back-reactions and a biphasic model (SFORB) for
+metabolites were present in `mkin` from the very beginning.
+
+## Derived software tools
+
+Soon after the publication of `mkin`, two derived tools were published, namely
+KinGUII (available from Bayer Crop Science) and CAKE (commissioned to Tessella
+by Syngenta), which added a graphical user interface (GUI), and added fitting by
+iteratively reweighted least squares (IRLS) and characterisation of likely
+parameter distributions by Markov Chain Monte Carlo (MCMC) sampling.
+
+CAKE focuses on a smooth use experience, sacrificing some flexibility in the model
+definition, originally allowing only two primary metabolites in parallel.
+The current version 3.3 of CAKE release in March 2016 uses a basic scheme for
+up to six metabolites in a flexible arrangement, but does not support
+back-reactions (non-instantaneous equilibria) or biphasic kinetics for metabolites.
+
+KinGUI offers an even more flexible widget for specifying complex kinetic
+models.  Back-reactions (non-instantaneous equilibria) were supported early on,
+but until 2014,  only simple first-order models could be specified for
+transformation products.  Starting with KinGUII version 2.1, biphasic modelling
+of metabolites was also available in KinGUII.
+
+A further graphical user interface (GUI) that has recently been brought to a decent
+degree of maturity is the browser based GUI named `gmkin`. Please see its
+[documentation page](https://pkgdown.jrwb.de/gmkin) and
+[manual](https://pkgdown.jrwb.de/gmkin/articles/gmkin_manual.html)
+for further information.
+
+A comparison of scope, usability and numerical results obtained with these
+tools has been recently been published by @ranke2018.
+
+## Recent developments
+
+Currently (July 2019), the main features available in `mkin` which are not
+present in KinGUII or CAKE, are the speed increase by using compiled code when
+a compiler is present, parallel model fitting on multicore machines using the
+`mmkin` function, and the estimation of parameter confidence intervals based on
+transformed parameters.
+
+In addition, the possibility to use two alternative error models to constant
+variance have been integrated. The variance by variable error model introduced
+by @gao11 has been available via an iteratively reweighted least squares (IRLS)
+procedure since mkin
+[version 0.9-22](https://pkgdown.jrwb.de/mkin/news/index.html#mkin-0-9-22-2013-10-26).
+With [release 0.9.49.5](https://pkgdown.jrwb.de/mkin/news/index.html#mkin-0-9-49-5-2019-07-04),
+the IRLS algorithm has been replaced by direct or step-wise maximisation of
+the likelihood function, which makes it possible not only to fit the
+variance by variable error model but also a
+[two-component error model](https://pkgdown.jrwb.de/mkin/reference/sigma_twocomp.html)
+inspired by error models developed in analytical chemistry.
+
+# Internal parameter transformations
+
+For rate constants, the log
+transformation is used, as proposed by Bates and Watts [-@bates1988, p. 77,
+149]. Approximate intervals are constructed for the transformed rate
+constants [compare @bates1988, p. 135], *i.e.* for their logarithms.
+Confidence intervals for the rate constants are then obtained using the
+appropriate backtransformation using the exponential function.
+
+In the first version of `mkin` allowing for specifying models using
+formation fractions, a home-made reparameterisation was used in order to ensure
+that the sum of formation fractions would not exceed unity.
+
+This method is still used in the current version of KinGUII (v2.1 from April
+2014), with a modification that allows for fixing the pathway to sink to zero.
+CAKE uses penalties in the objective function in order to enforce this
+constraint.
+
+In 2012, an alternative reparameterisation of the formation fractions was
+proposed together with René Lehmann [@ranke2012], based on isometric
+logratio transformation (ILR). The aim was to improve the validity of the
+linear approximation of the objective function during the parameter
+estimation procedure as well as in the subsequent calculation of parameter
+confidence intervals.
+
+## Confidence intervals based on transformed parameters
+
+In the first attempt at providing improved parameter confidence intervals
+introduced to `mkin` in 2013, confidence intervals obtained from
+FME on the transformed parameters were simply all backtransformed one by one
+to yield asymmetric confidence intervals for the backtransformed parameters.
+
+However, while there is a 1:1 relation between the rate constants in the model
+and the transformed parameters fitted in the model, the parameters obtained by the
+isometric logratio transformation are calculated from the set of formation
+fractions that quantify the paths to each of the compounds formed from a
+specific parent compound, and no such 1:1 relation exists.
+
+Therefore, parameter confidence intervals for formation fractions obtained with
+this method only appear valid for the case of a single transformation product, where
+only one formation fraction is to be estimated, directly corresponding to one
+component of the ilr transformed parameter.
+
+The confidence intervals obtained by backtransformation for the cases where a
+1:1 relation between transformed and original parameter exist are considered by
+the author of this vignette to be more accurate than those obtained using a
+re-estimation of the Hessian matrix after backtransformation, as implemented
+in the FME package.
+
+## Parameter t-test based on untransformed parameters
+
+The standard output of many nonlinear regression software packages includes
+the results from a test for significant difference from zero for all parameters.
+Such a test is also recommended to check the validity of rate constants in the
+FOCUS guidance [@FOCUSkinetics2014, p.  96ff].
+
+It has been argued that the precondition for this test, *i.e.* normal distribution
+of the estimator for the parameters, is not fulfilled in the case of nonlinear regression
+[@ranke2015]. However, this test is commonly used by industry, consultants and
+national authorities in order to decide on the reliability of parameter estimates, based
+on the FOCUS guidance mentioned above. Therefore, the results of this one-sided
+t-test are included in the summary output from `mkin`.
+
+As it is not reasonable to test for significant difference of the transformed
+parameters (*e.g.* $log(k)$) from zero, the t-test is calculated based on the
+model definition before parameter transformation, *i.e.* in a similar way as in
+packages that do not apply such an internal parameter transformation.  A note
+is included in the `mkin` output, pointing to the fact that the t-test is based
+on the unjustified assumption of normal distribution of the parameter
+estimators.
+
+# References
+
+<!-- vim: set foldmethod=syntax: -->