fit_DoseResponseCurve: Fit a dose-response curve for luminescence data (Lx/Tx...

In Luminescence: Comprehensive Luminescence Dating Data Analysis

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

Description

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 = c("interpolation", "extrapolation", "alternate"),
  fit.method = c("SSE", "LIN", "QDR", "SSE OR LIN", "SSE+LIN", "DSE", "GOK", "OTOR",
    "OTORX"),
  fit.force_through_origin = FALSE,
  fit.weights = c("inverse_var", "inverse_std", "norm_inverse_std"),
  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 (optional). If these column names are used, then they can be passed in whatever order; otherwise columns are taken by position. If object is a list, the function is called on each of its elements. If fit.method = "OTORX" you have to provide the test dose in the same unit as the dose in a column called Test_Dose. The function searches explicitly for this column name. Only the first value will be used assuming a constant test dose over the measurement cycle.
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, SSE, ⁠SSE OR LIN⁠, SSE+LIN, DSE (not defined for extrapolation), GOK, OTOR and OTORX. See details.
fit.force_through_origin	logical (with default) allow to force the fitted function through the origin. For method = "DSE" the function will be fixed through the origin in either case, so this option will have no effect.
fit.weights	character numeric (with default): weighting approach to be used for the fitting. Options are inverse_var (default), inverse_std, norm_inverse_std, a numeric vector, or NULL (no weighting). If the input is a numeric vector, it must have length equal to the number of data points to fit (usually the LxTx values). 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 to use with the fit methods SSE, SSE+LIN, ⁠SSE OR LIN⁠, GOK, OTOR, OTORX Argument to be inserted for experimental application only!
n.MC	integer (with default): number of Monte Carlo simulations for error estimation.
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).

Details

Implemented fitting methods

For all options (except for the LIN, QDR and the ⁠SSE 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.

Keyword: LIN

Fits a linear function to the data using lm:

y = mx + D_i

Keyword: QDR

Fits a linear function with a quadratic term to the data using lm:

y = a + bx + cx^2

Keyword: SSE (formerly EXP)

Fits a single saturating exponential function of the form

y = N (1 - \exp(-\frac{x + D_i}{D_0}))

Parameters D_0 and D_i are approximated by a linear fit using lm.

Keyword: ⁠SSE OR LIN⁠ (formerly ⁠EXP OR LIN⁠)

Works for some cases where an SSE fit fails. If the SSE fit fails, a LIN fit is done instead, which always works.

Keyword: SSE+LIN (formerly EXP+LIN)

Tries to fit an exponential plus linear function of the form:

y = N(1 - \exp(-\frac{x + D_i}{D_0}) + gx)

The D_e is calculated by iteration.

Note: In the context of luminescence dating, this function has no physical meaning. Therefore, no D_0 value is returned.

Keyword: DSE (formerly EXP+EXP)

Tries to fit a double exponential function of the form

y = N_1 (1 - \exp(-\frac{x + D_i}{D0_1})) + N_2 (1 - \exp(-\frac{x + D_i}{D0_2}))

This fitting procedure is not really robust against wrong start parameters.

Keyword: GOK

Tries to fit the general-order kinetics function following Guralnik et al. (2015) of the form

y = a (d - (1 + \frac{1}{D_0} x c)^{-1 / c})

where c > 0 is a kinetic order modifier.

Keyword: OTOR (formerly LambertW)

This tries to fit a dose-response curve based on the Lambert W function and the one trap one recombination centre (OTOR) model according to Pagonis et al. (2020). The function has the form:

y = (1 + (\mathcal{W}((R - 1) * \exp(R - 1 - (x + D_i) / D_c)) / (1 - R))) * N

with W the Lambert-W function (calculated using lamW::lambertW0), R the dimensionless retrapping ratio, N the total concentration of trappings states in cm^{-3}, D_{c} = N/R a constant, and D_{i} is the offset on the x-axis (not part of the original formula in Pagonis et al. 2020). Note that R and D_{c} have a valid physical interpretation only when saturation is reached. 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.

Keyword: OTORX

This adapts extended OTOR (therefore: OTORX) model proposed by Lawless and Timar-Gabor (2024) accounting for retrapping (the equation implemented here is written slightly differently than in the original manuscript):

F_{OTORX} = 1 + \left[\mathcal{W}\left(-Q * \exp\left(-Q-(1-Q(1-\frac{1}{\exp(1)})) \frac{D + D_i}{D_{63}}\right)\right)\right] / Q

with

Q = \frac{A_m - A_n}{A_m}\frac{N}{N+N_D}

where A_m and A_n are rate constants for the recombination and the trapping of electrons (N), respectively. D_{63} corresponds to the value at which the trap occupation corresponds to 63% of the saturation value. D_i is an offset: if set to zero, the curve will be forced through the origin as in the original publication.

For the implementation the calculation reads further

y = \frac{F_{OTORX}(((D + D_i)/D_{63}), Q)}{F_{OTORX}((D_{test} + D_i)/D_{63}, Q)}

with D_{test} being the test dose in the same unit (usually s or Gy) as the regeneration dose points. This value is essential and needs to provided along with the usual dose and \frac{L_x}{T_x} values (see object parameter input and the example section). For more details see Lawless and Timar-Gabor (2024).

The fit also returns the parameter R know from OTOR, which is derived as R = 1 - Q.

Note: The offset adder D_i is not part of the formula in Timar-Gabor (2024) and can be set to zero with the option fit.force_through_origin = TRUE

Fit weighting

• "inverse_var" (inverse variance weighting - current default)

  w_i = \frac{1}{\sigma_i^2}

• "inverse_std" (inverse standard error)

  w_i = \frac{1}{\sigma_i}

• "norm_inverse_std" (normalised inverse standard error weighting - default up to v1.2.1)

  w_i = \frac{\frac{1}{\sigma_i}}{\Sigma{\frac{1}{\sigma_i}}}

  Although used until Luminescence v1.2.1, this method is no longer recommended, as it does not align with the mathematical approach used in common nls fitting methods.

If the option fit.weights = NULL all weights are set to 1, which disables weighting altogether. If fit.weights is a numeric vector of correct length (same number of rows as the input LxTx), then those fit weights are used. This may be helpful to compare different fitting algorithms that have implemented fit weights differently.

Error estimation using Monte Carlo simulation

Error estimation is done using a parametric bootstrap. A set of \frac{L_x}{T_x} values is constructed by randomly drawing curve data from normal distributions defined by the input values (mean = value, sd = value.error). A dose-response curve is then fitted for each sampled dataset using the chosen fitting method, producing a distribution of single De values. The standard deviation of this distribution is taken as the error of the De. With more iterations (n.MC) the error estimate stabilizes. 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 (SSE+LIN and DSE).

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
..$Fit :	nls or lm	object from the fitting for SSE, SSE+LIN and DSE. In case of a resulting linear fit when using LIN, QDR or ⁠SSE OR LIN⁠
..Fit.Args :	list	Arguments to the function
..$Formula :	expression	Fitting formula as R expression

The ⁠@info⁠ slot contains the following elements:

DATA.OBJECT	TYPE	DESCRIPTION
..$fit_message:	character	The fit message reported
..$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_0 value, curvature parameter of the exponential
D01.ERROR	numeric	standard error of the D_0 value
D02	numeric	2nd D_0 value, only for DSE
D02.ERROR	numeric	standard error for 2nd D_0; only for DSE
R	numeric	the material specific parameter R (only OTOR and OTORX)
R.LOWER	numeric	lower 25% quantile of R
R.UPPER	numeric	upper 75% quantile of R
Dc	numeric	value indicating saturation level; only for OTOR
Dc.LOWER	numeric	lower 25% quantile for Dc; only for OTOR
Dc.UPPER	numeric	upper 75% quantile for Dc; only for OTOR
D63	numeric	the specific saturation level; only for OTOR, OTORX
D63.LOWER \ tab numeric	lower 25% quantile of D63; only for OTOR, OTORX
D63.UPPER \ tab numeric	upper 75% quantile of D63; only for OTOR, OTORX
D80	numeric	the specific saturation level; only for SSE, OTOR, OTORX
D80.LOWER \ tab numeric	lower 25% quantile of D80; only for OTOR, OTORX
D80.UPPER \ tab numeric	upper 75% quantile of D80; only for OTOR, OTORX
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
Fit	character	applied fit function
Mode	character	mode used in fitting
HPDI68_L	numeric	highest probability density of the 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
.De.plot	numeric	equivalent dose used internally for plotting
.De.raw	numeric	equivalent dose reported 'as is', that is, containing infinities and negative values if they could be calculated. Bear in mind that negative values are meaningless and may be arbitrary.

Function version

1.7

How to cite

Kreutzer, S., Dietze, M., Colombo, M., 2026. fit_DoseResponseCurve(): Fit a dose-response curve for luminescence data (Lx/Tx against dose). Function version 1.7. 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., de Boer, A., Bluszcz, A., 2026. Luminescence: Comprehensive Luminescence Dating Data Analysis. R package version 1.3.1. https://r-lum.github.io/Luminescence/

Author(s)

Sebastian Kreutzer, F2.1 Geophysical Parametrisation/Regionalisation, LIAG - Institute for Applied Geophysics (Germany)
Michael Dietze, RWTH Aachen (Germany)
Marco Colombo, Institute of Geography, Heidelberg University (Germany) , RLum Developer Team

References

Berger, G.W., Huntley, D.J., 1989. Test data for exponential fits. Ancient TL 7, 43-46. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.26034/la.atl.1989.150")}

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.radmeas.2015.02.011")}

Lawless, J.L., Timar-Gabor, A., 2024. A new analytical model to fit both fine and coarse grained quartz luminescence dose response curves. Radiation Measurements 170, 107045. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.radmeas.2023.107045")}

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jlumin.2020.117333")}

See Also

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

Examples

##(1) fit growth curve for a dummy data.set and show De value
data(ExampleData.LxTxData, envir = environment())
temp <- fit_DoseResponseCurve(LxTxData)
get_RLum(temp)

##(1b) to access the fitting value try
get_RLum(temp, data.object = "Fit")

##(2) fit using the 'extrapolation' mode
LxTxData[1,2:3] <- c(0.5, 0.001)
print(fit_DoseResponseCurve(LxTxData, mode = "extrapolation"))

##(3) fit using the 'alternate' mode
LxTxData[1,2:3] <- c(0.5, 0.001)
print(fit_DoseResponseCurve(LxTxData, mode = "alternate"))

##(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)

#calculate confidence interval for the parameters
#as alternative error estimation
confint(results$Fit, level = 0.68)

## Not run: 
##(5) special case the OTORX model with test dose column
df <- cbind(LxTxData, Test_Dose = 15)
fit_DoseResponseCurve(object = df, fit.method = "OTORX", n.MC = 10) |>
 plot_DoseResponseCurve()

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)
 
## End(Not run)

Luminescence documentation built on Sept. 18, 2026, 9:07 a.m.