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Nonlinear Least Squares Fit for LM-OSL curves

Source: R/fit_LMCurve.R
fit_LMCurve.Rd

The function determines weighted nonlinear least-squares estimates of the component parameters of an LM-OSL curve (Bulur 1996) for a given number of components and returns various component parameters. The fitting procedure uses the function nls with the port algorithm.

Usage

fit_LMCurve(
  values,
  values.bg,
  n.components = 3,
  start_values,
  input.dataType = "LM",
  fit.method = "port",
  sample_code = "",
  sample_ID = "",
  LED.power = 36,
  LED.wavelength = 470,
  fit.trace = FALSE,
  fit.advanced = FALSE,
  fit.calcError = FALSE,
  bg.subtraction = "polynomial",
  verbose = TRUE,
  plot = TRUE,
  plot.BG = FALSE,
  ...
)

Arguments

values

RLum.Data.Curve or data.frame (required): x,y data of measured values (time and counts). See examples.

values.bg

RLum.Data.Curve or data.frame (optional): x,y data of measured values (time and counts) for background subtraction.

n.components

integer (with default): fixed number of components that are to be recognised during fitting (min = 1, max = 7).

start_values

data.frame (optional): start parameters for lm and xm data for the fit. If no start values are given, an automatic start value estimation is attempted (see details).

input.dataType

character (with default): alter the plot output depending on the input data: "LM" or "pLM" (pseudo-LM). See: CW2pLM

fit.method

character (with default): select fit method, allowed values: 'port' and 'LM'. 'port' uses the 'port' routine from the function nls 'LM' utilises the function nlsLM from the package minpack.lm and with that the Levenberg-Marquardt algorithm.

sample_code

character (optional): sample code used for the plot and the optional output table (mtext).

sample_ID

character (optional): additional identifier used as column header for the table output.

LED.power

numeric (with default): LED power (max.) used for intensity ramping in mW/cm^2. Note: This value is used for the calculation of the absolute photoionisation cross section.

LED.wavelength

numeric (with default): LED wavelength in nm used for stimulation. Note: This value is used for the calculation of the absolute photoionisation cross section.

fit.trace

logical (with default): traces the fitting process on the terminal.

fit.advanced

logical (with default): enables advanced fitting attempt for automatic start parameter recognition. Works only if no start parameters are provided. Note: It may take a while and it is not compatible with fit.method = "LM".

fit.calcError

logical (with default): calculate 1-sigma error range of components using stats::confint.

bg.subtraction

character (with default): specifies method for background subtraction (polynomial, linear, channel, see Details). Note: requires input for values.bg.

verbose

logical (with default): terminal output with fitting results.

plot

logical (with default): returns a plot of the fitted curves.

plot.BG

logical (with default): returns a plot of the background values with the fit used for the background subtraction.

...

Further arguments that may be passed to the plot output, e.g. xlab, xlab, main, log.

Value

Various types of plots are returned. For details see above. Furthermore an RLum.Results object is returned with the following structure:

@data:

.. $data : data.frame with fitting results
.. $fit : nls (nls object)
.. $component_matrix : matrix with numerical xy-values of the single fitted components with the resolution of the input data .. $component.contribution.matrix : list component distribution matrix

info:

.. $call : call the original function call

Matrix structure for the distribution matrix:

Column 1 and 2: time and rev(time) values
Additional columns are used for the components, two for each component, containing I0 and n0. The last columns cont. provide information on the relative component contribution for each time interval including the row sum for this values.

Details

Fitting function

The function for the fitting has the general form:

$$y = (exp(0.5)*Im_1*x/xm_1)*exp(-x^2/(2*xm_1^2)) + ,\ldots, + exp(0.5)*Im_i*x/xm_i)*exp(-x^2/(2*xm_i^2))$$

where \(1 < i < 8\)

This function and the equations for the conversion to b (detrapping probability) and n0 (proportional to initially trapped charge) have been taken from Kitis et al. (2008):

$$xm_i=\sqrt{max(t)/b_i}$$ $$Im_i=exp(-0.5)n0/xm_i$$

Background subtraction

Three methods for background subtraction are provided for a given background signal (values.bg).

  • polynomial: default method. A polynomial function is fitted using glm and the resulting function is used for background subtraction: $$y = a*x^4 + b*x^3 + c*x^2 + d*x + e$$

  • linear: a linear function is fitted using glm and the resulting function is used for background subtraction: $$y = a*x + b$$

  • channel: the measured background signal is subtracted channel wise from the measured signal.

Start values

The choice of the initial parameters for the nls-fitting is a crucial point and the fitting procedure may mainly fail due to ill chosen start parameters. Here, three options are provided:

(a) If no start values (start_values) are provided by the user, a cheap guess is made by using the detrapping values found by Jain et al. (2003) for quartz for a maximum of 7 components. Based on these values, the pseudo start parameters xm and Im are recalculated for the given data set. In all cases, the fitting starts with the ultra-fast component and (depending on n.components) steps through the following values. If no fit could be achieved, an error plot (for plot = TRUE) with the pseudo curve (based on the pseudo start parameters) is provided. This may give the opportunity to identify appropriate start parameters visually.

(b) If start values are provided, the function works like a simple nls fitting approach.

(c) If no start parameters are provided and the option fit.advanced = TRUE is chosen, an advanced start parameter estimation is applied using a stochastic attempt. Therefore, the recalculated start parameters (a) are used to construct a normal distribution. The start parameters are then sampled randomly from this distribution. A maximum of 100 attempts will be made. Note: This process may be time consuming.

Goodness of fit

The goodness of the fit is given by a pseudo-R^2 value (pseudo coefficient of determination). According to Lave (1970), the value is calculated as:

$$pseudoR^2 = 1 - RSS/TSS$$

where \(RSS = Residual~Sum~of~Squares\) and \(TSS = Total~Sum~of~Squares\)

Error of fitted component parameters

The 1-sigma error for the components is calculated using the function stats::confint. Due to considerable calculation time, this option is deactivated by default. In addition, the error for the components can be estimated by using internal R functions like summary. See the nls help page for more information.

For more details on the nonlinear regression in R, see Ritz & Streibig (2008).

Note

The pseudo-R^2 may not be the best parameter to describe the goodness of the fit. The trade off between the n.components and the pseudo-R^2 value currently remains unconsidered.

The function does not ensure that the fitting procedure has reached a global minimum rather than a local minimum! In any case of doubt, the use of manual start values is highly recommended.

Function version

0.3.4

How to cite

Kreutzer, S., 2024. fit_LMCurve(): Nonlinear Least Squares Fit for LM-OSL curves. Function version 0.3.4. 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., 2024. Luminescence: Comprehensive Luminescence Dating Data Analysis. R package version 0.9.26. https://r-lum.github.io/Luminescence/

References

Bulur, E., 1996. An Alternative Technique For Optically Stimulated Luminescence (OSL) Experiment. Radiation Measurements, 26, 5, 701-709.

Jain, M., Murray, A.S., Boetter-Jensen, L., 2003. Characterisation of blue-light stimulated luminescence components in different quartz samples: implications for dose measurement. Radiation Measurements, 37 (4-5), 441-449.

Kitis, G. & Pagonis, V., 2008. Computerized curve deconvolution analysis for LM-OSL. Radiation Measurements, 43, 737-741.

Lave, C.A.T., 1970. The Demand for Urban Mass Transportation. The Review of Economics and Statistics, 52 (3), 320-323.

Ritz, C. & Streibig, J.C., 2008. Nonlinear Regression with R. R. Gentleman, K. Hornik, & G. Parmigiani, eds., Springer, p. 150.

See also

fit_CWCurve, plot, nls, minpack.lm::nlsLM, get_RLum

Author

Sebastian Kreutzer, Institute of Geography, Heidelberg University (Germany) , RLum Developer Team

Examples


##(1) fit LM data without background subtraction
data(ExampleData.FittingLM, envir = environment())
fit_LMCurve(values = values.curve, n.components = 3, log = "x")
#> 
#> [fit_LMCurve()]
#> 
#> Fitting was done using a 3-component function:
#> 
#>        xm1        xm2        xm3        Im1        Im2        Im3 
#>   56.18298 1449.72480 7878.26800  202.76619  367.30236  639.21452 
#> 
#> (equation used for fitting according to Kitis & Pagonis, 2008)
#> ------------------------------------------------------------------------------
#> (1) Corresponding values according to the equation in Bulur, 1996 for b and n0:
#> 
#> b1 = 1.267216e+00 +/- NA
#> n01 = 1.878225e+04 +/- NA
#> 
#> b2 = 1.903219e-03 +/- NA
#> n02 = 8.779232e+05 +/- NA
#> 
#> b3 = 6.444637e-05 +/- NA
#> n03 = 8.302801e+06 +/- NA
#> 
#> cs from component.1 = 1.488e-17 cm^2	 >> relative: 1
#> cs from component.2 = 2.234e-20 cm^2	 >> relative: 0.0015
#> cs from component.3 = 7.566e-22 cm^2	 >> relative: 1e-04
#> 
#> (stimulation intensity value used for calculation: 8.517726e+16 1/s 1/cm^2)
#> (errors quoted as 1-sigma uncertainties)
#> ------------------------------------------------------------------------------
#> 
#> pseudo-R^2 = 0.9557


##(2) fit LM data with background subtraction and export as JPEG
## -alter file path for your preferred system
##jpeg(file = "~/Desktop/Fit_Output\%03d.jpg", quality = 100,
## height = 3000, width = 3000, res = 300)
data(ExampleData.FittingLM, envir = environment())
fit_LMCurve(values = values.curve, values.bg = values.curveBG,
            n.components = 2, log = "x", plot.BG = TRUE)
#> [fit_LMCurve] >> Background subtracted (method="polynomial")!

#> 
#> [fit_LMCurve()]
#> 
#> Fitting was done using a 2-component function:
#> 
#>        xm1        xm2        Im1        Im2 
#>   53.32087 1587.57202  176.74382  406.89925 
#> 
#> (equation used for fitting according to Kitis & Pagonis, 2008)
#> ------------------------------------------------------------------------------
#> (1) Corresponding values according to the equation in Bulur, 1996 for b and n0:
#> 
#> b1 = 1.406908e+00 +/- NA
#> n01 = 1.553777e+04 +/- NA
#> 
#> b2 = 1.587059e-03 +/- NA
#> n02 = 1.065044e+06 +/- NA
#> 
#> cs from component.1 = 1.652e-17 cm^2	 >> relative: 1
#> cs from component.2 = 1.863e-20 cm^2	 >> relative: 0.0011
#> 
#> (stimulation intensity value used for calculation: 8.517726e+16 1/s 1/cm^2)
#> (errors quoted as 1-sigma uncertainties)
#> ------------------------------------------------------------------------------
#> 
#> pseudo-R^2 = 0.9417

##dev.off()

##(3) fit LM data with manual start parameters
data(ExampleData.FittingLM, envir = environment())
fit_LMCurve(values = values.curve,
            values.bg = values.curveBG,
            n.components = 3,
            log = "x",
            start_values = data.frame(Im = c(170,25,400), xm = c(56,200,1500)))
#> [fit_LMCurve] >> Background subtracted (method="polynomial")!
#> 
#> [fit_LMCurve()]
#> 
#> Fitting was done using a 3-component function:
#> 
#>        xm1        xm2        xm3        Im1        Im2        Im3 
#>   49.00634  204.39749 1591.66438  169.43979   23.00769  405.46174 
#> 
#> (equation used for fitting according to Kitis & Pagonis, 2008)
#> ------------------------------------------------------------------------------
#> (1) Corresponding values according to the equation in Bulur, 1996 for b and n0:
#> 
#> b1 = 1.665542e+00 +/- NA
#> n01 = 1.369036e+04 +/- NA
#> 
#> b2 = 9.574341e-02 +/- NA
#> n02 = 7.753465e+03 +/- NA
#> 
#> b3 = 1.578909e-03 +/- NA
#> n03 = 1.064017e+06 +/- NA
#> 
#> cs from component.1 = 1.955e-17 cm^2	 >> relative: 1
#> cs from component.2 = 1.124e-18 cm^2	 >> relative: 0.0575
#> cs from component.3 = 1.854e-20 cm^2	 >> relative: 9e-04
#> 
#> (stimulation intensity value used for calculation: 8.517726e+16 1/s 1/cm^2)
#> (errors quoted as 1-sigma uncertainties)
#> ------------------------------------------------------------------------------
#> 
#> pseudo-R^2 = 0.9437