Fit a dose-response curve for luminescence data (Lx/Tx against dose)

A dose-response curve is produced for luminescence measurements using a regenerative or additive protocol. The function supports interpolation and extrapolation to calculate the equivalent dose.

Usage

fit_DoseResponseCurve(
  object,
  mode = "interpolation",
  fit.method = "EXP",
  fit.force_through_origin = FALSE,
  fit.weights = TRUE,
  fit.includingRepeatedRegPoints = TRUE,
  fit.NumberRegPoints = NULL,
  fit.NumberRegPointsReal = NULL,
  fit.bounds = TRUE,
  n.MC = 100,
  txtProgressBar = TRUE,
  verbose = TRUE,
  ...
)

Arguments

object

data.frame or a list of such objects (required): data frame with columns for Dose, LxTx, LxTx.Error and TnTx. The column for the test dose response is optional, but requires 'TnTx' as column name if used. For exponential fits at least three dose points (including the natural) should be provided. If object is a list, the function is called on each of its elements.

mode

character (with default): selects calculation mode of the function.

• "interpolation" (default) calculates the De by interpolation,

• "extrapolation" calculates the equivalent dose by extrapolation (useful for MAAD measurements) and

• "alternate" calculates no equivalent dose and just fits the data points.

Please note that for option "interpolation" the first point is considered as natural dose

fit.method

character (with default): function used for fitting. Possible options are:

• LIN,

• QDR,

• EXP,

• EXP OR LIN,

• EXP+LIN,

• EXP+EXP,

• GOK,

• LambertW

See details.

fit.force_through_origin

logical (with default) allow to force the fitted function through the origin. For method = "EXP+EXP" the function will be fixed through the origin in either case, so this option will have no effect.

fit.weights

logical (with default): option whether the fitting is done with or without weights. See details.

fit.includingRepeatedRegPoints

logical (with default): includes repeated points for fitting (TRUE/FALSE).

fit.NumberRegPoints

integer (optional): set number of regeneration points manually. By default the number of all (!) regeneration points is used automatically.

fit.NumberRegPointsReal

integer (optional): if the number of regeneration points is provided manually, the value of the real, regeneration points = all points (repeated points) including reg 0, has to be inserted.

fit.bounds

logical (with default): set lower fit bounds for all fitting parameters to 0. Limited for the use with the fit methods EXP, EXP+LIN, EXP OR LIN, GOK, LambertW Argument to be inserted for experimental application only!

n.MC

integer (with default): number of Monte Carlo simulations for error estimation, see details.

txtProgressBar

logical (with default): enable/disable the progress bar. If verbose = FALSE also no txtProgressBar is shown.

verbose

logical (with default): enable/disable output to the terminal.

...

Further arguments to be passed (currently ignored).

Value

An RLum.Results object is returned containing the slot data with the following elements:

Overview elements

DATA.OBJECT	TYPE	DESCRIPTION
..$De :	data.frame	Table with De values
..$De.MC :	numeric	Table with De values from MC runs
..$Fit :	nls or lm	object from the fitting for EXP, EXP+LIN and EXP+EXP. In case of a resulting linear fit when using LIN, QDR or EXP OR LIN
..Fit.Args :	list	Arguments to the function
..$Formula :	expression	Fitting formula as R expression
..$call :	call	The original function call

If object is a list, then the function returns a list of RLum.Results objects as defined above.

Details - DATA.OBJECT$De This object is a data.frame with the following columns

De	numeric	equivalent dose
De.Error	numeric	standard error the equivalent dose
D01	numeric	D-naught value, curvature parameter of the exponential
D01.ERROR	numeric	standard error of the D-naught value
D02	numeric	2nd D-naught value, only for EXP+EXP
D02.ERROR	numeric	standard error for 2nd D-naught; only for EXP+EXP
Dc	numeric	value indicating saturation level; only for LambertW
n_N	numeric	saturation level of dose-response curve derived via integration from the used function; it compares the full integral of the curves (N) to the integral until De (n) (e.g., Guralnik et al., 2015)
De.MC	numeric	equivalent dose derived by Monte-Carlo simulation; ideally identical to De
De.plot	numeric	equivalent dose use for plotting
Fig	character	applied fit function
HPDI68_L	numeric	highest probability density of approximated equivalent dose probability curve representing the lower boundary of 68% probability
HPDI68_U	numeric	same as HPDI68_L for the upper bound
HPDI95_L	numeric	same as HPDI68_L but for 95% probability
HPDI95_U	numeric	same as HPDI95_L but for the upper bound

Details

Fitting methods

For all options (except for the LIN, QDR and the EXP OR LIN), the minpack.lm::nlsLM function with the LM (Levenberg-Marquardt algorithm) algorithm is used. Note: For historical reasons for the Monte Carlo simulations partly the function nls using the port algorithm.

The solution is found by transforming the function or using stats::uniroot.

LIN: fits a linear function to the data using lm: $$y = mx + n$$

QDR: fits a linear function with a quadratic term to the data using lm: $$y = a + bx + cx^2$$

EXP: tries to fit a function of the form $$y = a(1 - exp(-\frac{(x+c)}{b}))$$ Parameters b and c are approximated by a linear fit using lm. Note: b = D0

EXP OR LIN: works for some cases where an EXP fit fails. If the EXP fit fails, a LIN fit is done instead.

EXP+LIN: tries to fit an exponential plus linear function of the form: $$y = a(1-exp(-\frac{x+c}{b}) + (gx))$$ The \(D_e\) is calculated by iteration.

Note: In the context of luminescence dating, this function has no physical meaning. Therefore, no D0 value is returned.

EXP+EXP: tries to fit a double exponential function of the form $$y = (a_1 (1-exp(-\frac{x}{b_1}))) + (a_2 (1 - exp(-\frac{x}{b_2})))$$ This fitting procedure is not robust against wrong start parameters and should be further improved.

GOK: tries to fit the general-order kinetics function after Guralnik et al. (2015) of the form of

$$y = a (d - (1 + (\frac{1}{b}) x c)^{(-1/c)})$$

where c > 0 is a kinetic order modifier (not to be confused with c in EXP or EXP+LIN!).

LambertW: tries to fit a dose-response curve based on the Lambert W function according to Pagonis et al. (2020). The function has the form

$$y ~ (1 + (W((R - 1) * exp(R - 1 - ((x + D_{int}) / D_{c}))) / (1 - R))) * N$$

with \(W\) the Lambert W function, calculated using the package lamW::lambertW0, \(R\) the dimensionless retrapping ratio, \(N\) the total concentration of trappings states in cm^-3 and \(D_{c} = N/R\) a constant. \(D_{int}\) is the offset on the x-axis. Please note that finding the root in mode = "extrapolation" is a non-easy task due to the shape of the function and the results might be unexpected.

Fit weighting

If the option fit.weights = TRUE is chosen, weights are calculated using provided signal errors (Lx/Tx error): $$fit.weights = \frac{\frac{1}{error}}{\Sigma{\frac{1}{error}}}$$

Error estimation using Monte Carlo simulation

Error estimation is done using a parametric bootstrapping approach. A set of Lx/Tx values is constructed by randomly drawing curve data sampled from normal distributions. The normal distribution is defined by the input values (mean = value, sd = value.error). Then, a dose-response curve fit is attempted for each dataset resulting in a new distribution of single De values. The standard deviation of this distribution becomes then the error of the De. With increasing iterations, the error value becomes more stable. However, naturally the error will not decrease with more MC runs.

Alternatively, the function returns highest probability density interval estimates as output, users may find more useful under certain circumstances.

Note: It may take some calculation time with increasing MC runs, especially for the composed functions (EXP+LIN and EXP+EXP).
Each error estimation is done with the function of the chosen fitting method.

Function version

1.2.1

How to cite

Kreutzer, S., Dietze, M., Colombo, M., 2025. fit_DoseResponseCurve(): Fit a dose-response curve for luminescence data (Lx/Tx against dose). Function version 1.2.1. In: Kreutzer, S., Burow, C., Dietze, M., Fuchs, M.C., Schmidt, C., Fischer, M., Friedrich, J., Mercier, N., Philippe, A., Riedesel, S., Autzen, M., Mittelstrass, D., Gray, H.J., Galharret, J., Colombo, M., Steinbuch, L., Boer, A.d., 2025. Luminescence: Comprehensive Luminescence Dating Data Analysis. R package version 1.0.1. https://r-lum.github.io/Luminescence/

References

Berger, G.W., Huntley, D.J., 1989. Test data for exponential fits. Ancient TL 7, 43-46.

Guralnik, B., Li, B., Jain, M., Chen, R., Paris, R.B., Murray, A.S., Li, S.-H., Pagonis, P., Herman, F., 2015. Radiation-induced growth and isothermal decay of infrared-stimulated luminescence from feldspar. Radiation Measurements 81, 224-231.

Pagonis, V., Kitis, G., Chen, R., 2020. A new analytical equation for the dose response of dosimetric materials, based on the Lambert W function. Journal of Luminescence 225, 117333. doi:10.1016/j.jlumin.2020.117333

See also

plot_GrowthCurve, nls, RLum.Results, get_RLum, minpack.lm::nlsLM, lm, uniroot, lamW::lambertW0

Author

Sebastian Kreutzer, Institute of Geography, Heidelberg University (Germany)
Michael Dietze, GFZ Potsdam (Germany)
Marco Colombo, Institute of Geography, Heidelberg University (Germany) , RLum Developer Team

Examples

##(1) fit growth curve for a dummy data.set and show De value
data(ExampleData.LxTxData, envir = environment())
temp <- fit_DoseResponseCurve(LxTxData)
#> [fit_DoseResponseCurve()] Fit: EXP (interpolation) | De = 1737.88 | D01 = 1766.07
get_RLum(temp)
#>         De De.Error      D01 D01.ERROR D02 D02.ERROR Dc       n_N    De.MC
#> 1 1737.881 61.81878 1766.074  91.29813  NA        NA NA 0.5271352 1745.224
#>    De.plot Fit HPDI68_L HPDI68_U HPDI95_L HPDI95_U
#> 1 1737.881 EXP 1682.884 1815.013 1615.168  1868.69

##(1b) to access the fitting value try
get_RLum(temp, data.object = "Fit")
#> Nonlinear regression model
#>   model: y ~ a * (1 - exp(-(x + c)/b))
#>    data: data
#>        a        b        c 
#>    6.806 1766.074    5.051 
#>  weighted residual sum-of-squares: 0.0004268
#> 
#> Number of iterations to convergence: 4 
#> Achieved convergence tolerance: 1.49e-08

##(2) fit using the 'extrapolation' mode
LxTxData[1,2:3] <- c(0.5, 0.001)
print(fit_DoseResponseCurve(LxTxData, mode = "extrapolation"))
#> [fit_DoseResponseCurve()] Fit: EXP (extrapolation) | De = 109.74 | D01 = 2624.06
#> 
#>  [RLum.Results-class]
#> 	 originator: fit_DoseResponseCurve()
#> 	 data: 5
#>  	 .. $De : data.frame
#> 	 .. $De.MC : numeric
#> 	 .. $Fit : nls
#> 	 .. $Fit.Args : list
#> 	 .. $Formula : expression
#> 	 additional info elements:  1 

##(3) fit using the 'alternate' mode
LxTxData[1,2:3] <- c(0.5, 0.001)
print(fit_DoseResponseCurve(LxTxData, mode = "alternate"))
#> 
#>  [RLum.Results-class]
#> 	 originator: fit_DoseResponseCurve()
#> 	 data: 5
#>  	 .. $De : data.frame
#> 	 .. $De.MC : logical
#> 	 .. $Fit : nls
#> 	 .. $Fit.Args : list
#> 	 .. $Formula : expression
#> 	 additional info elements:  1 

##(4) import and fit test data set by Berger & Huntley 1989
QNL84_2_unbleached <-
read.table(system.file("extdata/QNL84_2_unbleached.txt", package = "Luminescence"))

results <- fit_DoseResponseCurve(
 QNL84_2_unbleached,
 mode = "extrapolation",
 verbose = FALSE)
#> Warning: [fit_DoseResponseCurve()] Error column invalid or 0, 'fit.weights' ignored

#calculate confidence interval for the parameters
#as alternative error estimation
confint(results$Fit, level = 0.68)
#> Waiting for profiling to be done...
#>           16%         84%
#> a 140543.3024 146731.8471
#> b    374.0861    425.5679
#> c    116.3499    133.3474


if (FALSE) { # \dontrun{
QNL84_2_bleached <-
read.table(system.file("extdata/QNL84_2_bleached.txt", package = "Luminescence"))
STRB87_1_unbleached <-
read.table(system.file("extdata/STRB87_1_unbleached.txt", package = "Luminescence"))
STRB87_1_bleached <-
read.table(system.file("extdata/STRB87_1_bleached.txt", package = "Luminescence"))

print(
 fit_DoseResponseCurve(
 QNL84_2_bleached,
 mode = "alternate",
 verbose = FALSE)$Fit)

print(
 fit_DoseResponseCurve(
 STRB87_1_unbleached,
 mode = "alternate",
 verbose = FALSE)$Fit)

print(
 fit_DoseResponseCurve(
 STRB87_1_bleached,
 mode = "alternate",
 verbose = FALSE)$Fit)
 } # }