Fit a kinetic model to data with one or more state variables

This function uses the Flexible Modelling Environment package FME to create a function calculating the model cost, i.e. the deviation between the kinetic model and the observed data. This model cost is then minimised using the Port algorithm nlminb, using the specified initial or fixed parameters and starting values. Per default, parameters in the kinetic models are internally transformed in order to better satisfy the assumption of a normal distribution of their estimators. In each step of the optimsation, the kinetic model is solved using the function mkinpredict. The variance of the residuals for each observed variable can optionally be iteratively reweighted until convergence using the argument reweight.method = "obs".

mkinfit(mkinmod, observed,
  parms.ini = "auto",
  state.ini = "auto",
  fixed_parms = NULL, fixed_initials = names(mkinmod$diffs)[-1],
  from_max_mean = FALSE,
  solution_type = c("auto", "analytical", "eigen", "deSolve"),
  method.ode = "lsoda",
  use_compiled = "auto",
  method.modFit = c("Port", "Marq", "SANN", "Nelder-Mead", "BFGS", "CG", "L-BFGS-B"),
  maxit.modFit = "auto",
  control.modFit = list(),
  transform_rates = TRUE,
  transform_fractions = TRUE,
  plot = FALSE, quiet = FALSE, err = NULL,
  weight = c("none", "std", "mean", "tc"),
  tc = c(sigma_low = 0.5, rsd_high = 0.07),
  scaleVar = FALSE,
  atol = 1e-8, rtol = 1e-10, n.outtimes = 100,
  reweight.method = NULL,
  reweight.tol = 1e-8, reweight.max.iter = 10,
  trace_parms = FALSE, ...)

Arguments

mkinmod	A list of class mkinmod, containing the kinetic model to be fitted to the data, or one of the shorthand names ("SFO", "FOMC", "DFOP", "HS", "SFORB"). If a shorthand name is given, a parent only degradation model is generated for the variable with the highest value in observed.
observed	The observed data. It has to be in the long format as described in modFit, i.e. the first column called "name" must contain the name of the observed variable for each data point. The second column must contain the times of observation, named "time". The third column must be named "value" and contain the observed values. Optionally, a further column can contain weights for each data point. Its name must be passed as a further argument named err which is then passed on to modFit.
parms.ini	A named vector of initial values for the parameters, including parameters to be optimised and potentially also fixed parameters as indicated by fixed_parms. If set to "auto", initial values for rate constants are set to default values. Using parameter names that are not in the model gives an error. It is possible to only specify a subset of the parameters that the model needs. You can use the parameter lists "bparms.ode" from a previously fitted model, which contains the differential equation parameters from this model. This works nicely if the models are nested. An example is given below.
state.ini	A named vector of initial values for the state variables of the model. In case the observed variables are represented by more than one model variable, the names will differ from the names of the observed variables (see map component of mkinmod). The default is to set the initial value of the first model variable to the mean of the time zero values for the variable with the maximum observed value, and all others to 0. If this variable has no time zero observations, its initial value is set to 100.
fixed_parms	The names of parameters that should not be optimised but rather kept at the values specified in parms.ini.
fixed_initials	The names of model variables for which the initial state at time 0 should be excluded from the optimisation. Defaults to all state variables except for the first one.
from_max_mean	If this is set to TRUE, and the model has only one observed variable, then data before the time of the maximum observed value (after averaging for each sampling time) are discarded, and this time is subtracted from all remaining time values, so the time of the maximum observed mean value is the new time zero.
solution_type	If set to "eigen", the solution of the system of differential equations is based on the spectral decomposition of the coefficient matrix in cases that this is possible. If set to "deSolve", a numerical ode solver from package deSolve is used. If set to "analytical", an analytical solution of the model is used. This is only implemented for simple degradation experiments with only one state variable, i.e. with no metabolites. The default is "auto", which uses "analytical" if possible, otherwise "eigen" if the model can be expressed using eigenvalues and eigenvectors, and finally "deSolve" for the remaining models (time dependence of degradation rates and metabolites). This argument is passed on to the helper function mkinpredict.
method.ode	The solution method passed via mkinpredict to ode in case the solution type is "deSolve". The default "lsoda" is performant, but sometimes fails to converge.
use_compiled	If set to FALSE, no compiled version of the mkinmod model is used, in the calls to mkinpredict even if a compiled verion is present.
method.modFit	The optimisation method passed to modFit. In order to optimally deal with problems where local minima occur, the "Port" algorithm is now used per default as it is less prone to get trapped in local minima and depends less on starting values for parameters than the Levenberg Marquardt variant selected by "Marq". However, "Port" needs more iterations. The former default "Marq" is the Levenberg Marquardt algorithm nls.lm from the package minpack.lm and usually needs the least number of iterations. The "Pseudo" algorithm is not included because it needs finite parameter bounds which are currently not supported. The "Newton" algorithm is not included because its number of iterations can not be controlled by control.modFit and it does not appear to provide advantages over the other algorithms.
maxit.modFit	Maximum number of iterations in the optimisation. If not "auto", this will be passed to the method called by modFit, overriding what may be specified in the next argument control.modFit.
control.modFit	Additional arguments passed to the optimisation method used by modFit.
transform_rates	Boolean specifying if kinetic rate constants should be transformed in the model specification used in the fitting for better compliance with the assumption of normal distribution of the estimator. If TRUE, also alpha and beta parameters of the FOMC model are log-transformed, as well as k1 and k2 rate constants for the DFOP and HS models and the break point tb of the HS model. If FALSE, zero is used as a lower bound for the rates in the optimisation.
transform_fractions	Boolean specifying if formation fractions constants should be transformed in the model specification used in the fitting for better compliance with the assumption of normal distribution of the estimator. The default (TRUE) is to do transformations. If TRUE, the g parameter of the DFOP and HS models are also transformed, as they can also be seen as compositional data. The transformation used for these transformations is the ilr transformation.
plot	Should the observed values and the numerical solutions be plotted at each stage of the optimisation?
quiet	Suppress printing out the current model cost after each improvement?
err	either NULL, or the name of the column with the error estimates, used to weigh the residuals (see details of modCost); if NULL, then the residuals are not weighed.
weight	only if err=NULL: how to weight the residuals, one of "none", "std", "mean", see details of modCost, or "tc" for the two component error model of Rocke and Lorenzato.
tc	The two components of the Rocke and Lorenzato error model as used for (initial) weighting
scaleVar	Will be passed to modCost. Default is not to scale Variables according to the number of observations.
atol	Absolute error tolerance, passed to ode. Default is 1e-8, lower than in lsoda.
rtol	Absolute error tolerance, passed to ode. Default is 1e-10, much lower than in lsoda.
n.outtimes	The length of the dataseries that is produced by the model prediction function mkinpredict. This impacts the accuracy of the numerical solver if that is used (see solution_type argument. The default value is 100.
reweight.method	The method used for iteratively reweighting residuals, also known as iteratively reweighted least squares (IRLS). Default is NULL, i.e. no iterative weighting. The first reweighting method is called "obs", meaning that each observed variable is assumed to have its own variance. This variance is estimated from the fit (mean squared residuals) and used for weighting the residuals in each iteration until convergence of this estimate up to reweight.tol or up to the maximum number of iterations specified by reweight.max.iter. The second reweighting method is called "tc" (two-component error model). When using this method, the two components of the error model according to Rocke and Lorenzato (1995) are estimated from the fit and the resulting variances are used for weighting the residuals in each iteration until convergence of these components or up to the maximum number of iterations specified.
reweight.tol	Tolerance for convergence criterion for the variance components in IRLS fits.
reweight.max.iter	Maximum iterations in IRLS fits.
trace_parms	Should a trace of the parameter values be listed?
…	Further arguments that will be passed to modFit.

Value

A list with "mkinfit" and "modFit" in the class attribute. A summary can be obtained by summary.mkinfit.

See also

Plotting methods plot.mkinfit and mkinparplot.

Fitting of several models to several datasets in a single call to mmkin.

Note

The implementation of iteratively reweighted least squares is inspired by the work of the KinGUII team at Bayer Crop Science (Walter Schmitt and Zhenglei Gao). A similar implemention can also be found in CAKE 2.0, which is the other GUI derivative of mkin, sponsored by Syngenta.

Note

When using the "IORE" submodel for metabolites, fitting with "transform_rates = TRUE" (the default) often leads to failures of the numerical ODE solver. In this situation it may help to switch off the internal rate transformation.

Source

Rocke, David M. und Lorenzato, Stefan (1995) A two-component model for measurement error in analytical chemistry. Technometrics 37(2), 176-184.

Examples

# Use shorthand notation for parent only degradation
fit <- mkinfit("FOMC", FOCUS_2006_C, quiet = TRUE)
summary(fit)
#> mkin version used for fitting:    0.9.47.1 
#> R version used for fitting:       3.4.3 
#> Date of fit:     Fri Mar  9 23:08:20 2018 
#> Date of summary: Fri Mar  9 23:08:20 2018 
#> 
#> Equations:
#> d_parent/dt = - (alpha/beta) * 1/((time/beta) + 1) * parent
#> 
#> Model predictions using solution type analytical 
#> 
#> Fitted with method Port using 64 model solutions performed in 0.148 s
#> 
#> Weighting: none
#> 
#> Starting values for parameters to be optimised:
#>          value   type
#> parent_0  85.1  state
#> alpha      1.0 deparm
#> beta      10.0 deparm
#> 
#> Starting values for the transformed parameters actually optimised:
#>               value lower upper
#> parent_0  85.100000  -Inf   Inf
#> log_alpha  0.000000  -Inf   Inf
#> log_beta   2.302585  -Inf   Inf
#> 
#> Fixed parameter values:
#> None
#> 
#> Optimised, transformed parameters with symmetric confidence intervals:
#>           Estimate Std. Error    Lower   Upper
#> parent_0  85.87000     2.2460 80.38000 91.3700
#> log_alpha  0.05192     0.1605 -0.34080  0.4446
#> log_beta   0.65100     0.2801 -0.03452  1.3360
#> 
#> Parameter correlation:
#>           parent_0 log_alpha log_beta
#> parent_0    1.0000   -0.2033  -0.3624
#> log_alpha  -0.2033    1.0000   0.9547
#> log_beta   -0.3624    0.9547   1.0000
#> 
#> Residual standard error: 2.275 on 6 degrees of freedom
#> 
#> Backtransformed parameters:
#> Confidence intervals for internally transformed parameters are asymmetric.
#> t-test (unrealistically) based on the assumption of normal distribution
#> for estimators of untransformed parameters.
#>          Estimate t value    Pr(>t)   Lower  Upper
#> parent_0   85.870  38.230 1.069e-08 80.3800 91.370
#> alpha       1.053   6.231 3.953e-04  0.7112  1.560
#> beta        1.917   3.570 5.895e-03  0.9661  3.806
#> 
#> Chi2 error levels in percent:
#>          err.min n.optim df
#> All data   6.657       3  6
#> parent     6.657       3  6
#> 
#> Estimated disappearance times:
#>         DT50  DT90 DT50back
#> parent 1.785 15.15     4.56
#> 
#> Data:
#>  time variable observed predicted residual
#>     0   parent     85.1    85.875  -0.7749
#>     1   parent     57.9    55.191   2.7091
#>     3   parent     29.9    31.845  -1.9452
#>     7   parent     14.6    17.012  -2.4124
#>    14   parent      9.7     9.241   0.4590
#>    28   parent      6.6     4.754   1.8460
#>    63   parent      4.0     2.102   1.8977
#>    91   parent      3.9     1.441   2.4590
#>   119   parent      0.6     1.092  -0.4919

# One parent compound, one metabolite, both single first order.
# Use mkinsub for convenience in model formulation. Pathway to sink included per default.
SFO_SFO <- mkinmod(
  parent = mkinsub("SFO", "m1"),
  m1 = mkinsub("SFO"))
#> Successfully compiled differential equation model from auto-generated C code.
# Fit the model to the FOCUS example dataset D using defaults
print(system.time(fit <- mkinfit(SFO_SFO, FOCUS_2006_D,
                           solution_type = "eigen", quiet = TRUE)))
#>    user  system elapsed 
#>   0.839   0.000   0.840 
coef(fit)
#>          parent_0 log_k_parent_sink   log_k_parent_m1     log_k_m1_sink 
#>          99.59848          -3.03822          -2.98030          -5.24750 
endpoints(fit)
#> $ff
#> parent_sink   parent_m1     m1_sink 
#>    0.485524    0.514476    1.000000 
#> 
#> $SFORB
#> logical(0)
#> 
#> $distimes
#>              DT50      DT90
#> parent   7.022929  23.32967
#> m1     131.760712 437.69961
#> 
# NOT RUN {
# deSolve is slower when no C compiler (gcc) was available during model generation
print(system.time(fit.deSolve <- mkinfit(SFO_SFO, FOCUS_2006_D,
                           solution_type = "deSolve")))
coef(fit.deSolve)
endpoints(fit.deSolve)
# }

# Use stepwise fitting, using optimised parameters from parent only fit, FOMC
# NOT RUN {
FOMC_SFO <- mkinmod(
  parent = mkinsub("FOMC", "m1"),
  m1 = mkinsub("SFO"))
# Fit the model to the FOCUS example dataset D using defaults
fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_2006_D, quiet = TRUE)
# Use starting parameters from parent only FOMC fit
fit.FOMC = mkinfit("FOMC", FOCUS_2006_D, quiet = TRUE)
fit.FOMC_SFO <- mkinfit(FOMC_SFO, FOCUS_2006_D, quiet = TRUE,
  parms.ini = fit.FOMC$bparms.ode)

# Use stepwise fitting, using optimised parameters from parent only fit, SFORB
SFORB_SFO <- mkinmod(
  parent = list(type = "SFORB", to = "m1", sink = TRUE),
  m1 = list(type = "SFO"))
# Fit the model to the FOCUS example dataset D using defaults
fit.SFORB_SFO <- mkinfit(SFORB_SFO, FOCUS_2006_D, quiet = TRUE)
fit.SFORB_SFO.deSolve <- mkinfit(SFORB_SFO, FOCUS_2006_D, solution_type = "deSolve",
                                 quiet = TRUE)
# Use starting parameters from parent only SFORB fit (not really needed in this case)
fit.SFORB = mkinfit("SFORB", FOCUS_2006_D, quiet = TRUE)
fit.SFORB_SFO <- mkinfit(SFORB_SFO, FOCUS_2006_D, parms.ini = fit.SFORB$bparms.ode, quiet = TRUE)
# }

# NOT RUN {
# Weighted fits, including IRLS
SFO_SFO.ff <- mkinmod(parent = mkinsub("SFO", "m1"),
                      m1 = mkinsub("SFO"), use_of_ff = "max")
f.noweight <- mkinfit(SFO_SFO.ff, FOCUS_2006_D, quiet = TRUE)
summary(f.noweight)
f.irls <- mkinfit(SFO_SFO.ff, FOCUS_2006_D, reweight.method = "obs", quiet = TRUE)
summary(f.irls)
f.w.mean <- mkinfit(SFO_SFO.ff, FOCUS_2006_D, weight = "mean", quiet = TRUE)
summary(f.w.mean)
f.w.value <- mkinfit(SFO_SFO.ff, subset(FOCUS_2006_D, value != 0), err = "value",
                     quiet = TRUE)
summary(f.w.value)
# }

# NOT RUN {
# Manual weighting
dw <- FOCUS_2006_D
errors <- c(parent = 2, m1 = 1)
dw$err.man <- errors[FOCUS_2006_D$name]
f.w.man <- mkinfit(SFO_SFO.ff, dw, err = "err.man", quiet = TRUE)
summary(f.w.man)
f.w.man.irls <- mkinfit(SFO_SFO.ff, dw, err = "err.man", quiet = TRUE,
                       reweight.method = "obs")
summary(f.w.man.irls)
# }