Smoothing and Fitting of an EPR Spectrum by Splines

Smoothing of the EPR spectra by non-parametric fitting a smoothing spline, ss from npreg package, onto the experimental EPR spectrum/spectra.

Usage

smooth_EPR_Spec_by_npreg(
  data.spectr,
  B = "B_mT",
  B.unit = "mT",
  lineSpecs.form = "derivative",
  Intensity = "dIepr_over_dB",
  method = "BIC",
  output.vec = FALSE,
  ...
)

Arguments

data.spectr

Spectrum data frame object where the magnetic flux density (in mT or G or T) column can be labeled as Field or B_G and that of the derivative intensity as dIepr_over_dB or single integrated intensity like Integrated_Intensity (index column might be included as well).

B

Character string, pointing to magnetic flux density column of the EPR spectrum data frame data.spectr either in "millitesla"/"tesla" or in "gauss", that is B = "B_mT" (default) or B = "B_G"/B = "T" or B = "Bsim_G" to include simulated EPR spectra as well.

B.unit

Character string, denoting the magnetic flux density unit e.g. B.unit = "G" (gauss, default) or B.unit = "mT"/"T" (millitesla/tesla).

lineSpecs.form

Character string, describing either "derivative" (default) or "integrated" (i.e. "absorption" which can be used as well) line form of the analyzed EPR spectrum/data.

Intensity

Character string, pointing to intensity column if other than dIepr_over_dB name/label is used (e.g. for simulated spectra), default: Intesity = "dIepr_over_dB"

method

Character string, corresponding to method to fit an EPR spectrum by a smoothing spline from {npreg} package (see the method argument in ss). The following methods are available (they are shortly described in Nathaniel's E. Helwig Introduction or additional information may be found in References) =>

Method	Short Description
"GCV"	Generalized Cross-Validation
"OCV"	Ordinary Cross-Validation
"GACV"	Generalized Approximate Cross-Validation
"ACV"	Approximate Cross-Validation
"REML"	Restricted Maximum Likelihood
"ML"	Maximum Likelihood
"AIC"	Akaike’s Information Criterion
"BIC"	Bayesian Information Criterion (default)

output.vec

Logical, whether the function output have to be vectorized, i.e. only the vector of smoothed EPR intensity is provided. This is especially useful for the spectral (time) series EPR data/spectra, which can be handily processed by the group_by using the pipe operators (%>%). Default: output.vec = NULL.

...

additional arguments passed to the function (see also ss function).

Value

If output.vec = TRUE the output corresponds to vector of the smoothed EPR intensity (either derivative lineSpecs.form = "derivative" or integrated lineSpecs.form = "integrated"/"absorption") with the length of the original Intensity. Contrarily, if the output.vec is set to FALSE the following list is returned =>

df: Original data frame with the addition column, corresponding to smoothed EPR intensity (derivative or integrated).
plot: Plot object (list) EPR intensity vs. B with the experimental data and its corresponding smoothed relation performed by splines.
rss: Weighted (if the optional parameter w is defined) sum of residual squares.
degs.freedom: Equivalent degrees of freedom used.
fit: List with elements to characterize the spline fit (Details see fit value in ss function documentation).
sigma: Estimated error standard deviation.
aic: Akaike’s Information Criterion (if method = "AIC"). A negative number that has the largest modulus (deepest down in the negative territory) indicates the preferred model.
bic: Bayesian Information Criterion (if method = "BIC"). A negative number that has the largest modulus (deepest down in the negative territory) indicates the preferred model.
log.likehood: Likelihood logarithm if method = "REML"/"ML". Log likelihood value is a measure of goodness of fit for any model. The higher the value, the better the model.

Details

The EPR spectrum is fitted by splines which, by default, correspond to cubic Bernoulli polynomials like $$I_{\text{EPR}}(B) = B^3 - (3/2)\,B^2 + (1/2)\,B$$ where $I_{\text{EPR}}$ equals to general EPR intensity and $B$ is the magnetic flux density. This may be applied to both derivative or single integrated EPR spectrum forms. Also a higher/lower polynomial degree may be applied by the method (see the argument description) and by the m optional argument (see ... additional arguments) which is a penalty order (integer). For the above-mentioned cubic spline/polynomial m = 2. Linear polynomial corresponds to m = 1 and the highest quintic polynomial/spline is referred as m = 3: $$I_{\text{EPR}}(B) = B^5 - (5/2)\,B^4 + (5/3)\,B^3 - (1/6)\,B$$ If the optional argument bernoulli is set to FALSE then "classic" definition of a smoothing spline is produced. In such case the function estimate is a piece-wise polynomial function with pieces of degree 2m−1. Additional optional arguments from ss like knots definition, equivalent degrees of freedom, vector of weights...etc. can be applied as well.

References

Berry LN, Helwig NE (2021). “Cross-Validation, Information Theory, or Maximum Likelihood? A Comparison of Tuning Methods for Penalized Splines.” Stats, 4(3), 701–724, https://doi.org/10.3390/stats4030042.

Elezović N (2016). “Generalized Bernoulli Polynomials and Numbers, Revisited.” Mediterr. J. Math., 13(1), 141–151. ISSN 1660-5454, https://doi.org/10.1007/s00009-014-0498-7.

Weisstein EW (2023). “Bernoulli Polynomial”, https://mathworld.wolfram.com/BernoulliPolynomial.html, MathWorld–A Wolfram Web Resource.

Helwig NE (2022). "Non-parametric Regression via Smoothing Splines", R package version 1.0-9, https://cran.r-project.org/web/packages/npreg/npreg.pdf.

Examples

## loading the built-in package example
## time series EPR spectra:
triarylamine.decay.series.dsc.path <-
load_data_example(file =
        "Triarylamine_radCat_decay_series.DSC")
triarylamine.decay.series.asc.path <-
load_data_example(file =
        "Triarylamine_radCat_decay_series.zip")
unzip(triarylamine.decay.series.asc.path,
      exdir = tempdir()
      )
## loading the kinetics:
triarylamine.decay.series.data <-
  readEPR_Exp_Specs_kin(name.root =
    "Triarylamine_radCat_decay_series",
    dir_ASC = tempdir(),
    dir_dsc_par =
      system.file("extdata",
                  package = "eprscope")
   )
#
## select the first spectrum
triarylamine.decay.series.data1st <-
   triarylamine.decay.series.data$df %>%
     dplyr::filter(time_s ==
       triarylamine.decay.series.data$time[1])
#
## smoothing the 1st EPR spectrum with default
## arguments/parameters
triarylamine.1st.spec.smooth <-
  smooth_EPR_Spec_by_npreg(data.spectr =
    triarylamine.decay.series.data1st
    )
#
## plot preview
triarylamine.1st.spec.smooth$plot

#
## sum of residual squares preview
triarylamine.1st.spec.smooth$rss
#> [1] 5.2912786e-07
#
## estimated error standard deviation
triarylamine.1st.spec.smooth$sigma
#> [1] 1.5366456e-05
#
## smoothing of all EPR spectra in the series
## with default arguments using the data
## "pipe" ("%>%") `dplyr` processing
triarylamine.all.spec.smooth <-
  triarylamine.decay.series.data$df %>%
    dplyr::group_by(time_s) %>%
    dplyr::mutate(smoothed =
      smooth_EPR_Spec_by_npreg(
        dplyr::pick(B_mT,dIepr_over_dB),
        output.vec = TRUE
     )
  )
#
## data frame preview
head(triarylamine.all.spec.smooth)
#> # A tibble: 6 × 6
#> # Groups:   time_s [1]
#>   index    B_G time_s dIepr_over_dB   B_mT      smoothed
#>   <int>  <dbl>  <dbl>         <dbl>  <dbl>         <dbl>
#> 1     1 3390        6  0.000013629  339    -0.0000037400
#> 2     2 3390.1      6 -0.0000010134 339.01 -0.0000038701
#> 3     3 3390.2      6 -0.000019795  339.02 -0.0000040017
#> 4     4 3390.2      6 -0.000029827  339.02 -0.0000041361
#> 5     5 3390.3      6 -0.000016871  339.03 -0.0000042745
#> 6     6 3390.4      6  0.0000025629 339.04 -0.0000044176
#
## plot all smoothed spectra in the series
plot_EPR_Specs(data.spectra =
          triarylamine.all.spec.smooth,
  Intensity = "smoothed",
  var2nd.series = "time_s",
  var2nd.series.slct.by = 10,
  line.colors = c("darkorange",
                  "darkblue"),
  legend.title = "Time (s)",
  yTicks = TRUE
  )