Extended Multiplicative Signal Correction

This function implements extended multiplicative signal correction (EMSC), as proposed by Martens and Stark (1991). EMSC extends msc() by modeling, in addition to the multiplicative and additive effects, a smooth polynomial baseline and, optionally, known interferent spectra.

Usage

emsc(
  x,
  xref = NULL,
  degree = 2,
  interferents = NULL,
  wavelength = NULL,
  robust = TRUE
)

Arguments

x: A numeric matrix or data frame, with one spectrum per row.
xref: An optional numeric vector giving the reference spectrum. If NULL (default), the median (robust = TRUE) or mean spectrum of x is used.
degree: A non-negative integer giving the degree of the polynomial baseline. Default is 2. With 0, only a constant offset is modeled.
interferents: An optional numeric vector (a single spectrum), matrix or data frame of interferent spectra, one per row, with the same number of columns as x.
wavelength: An optional numeric vector of wavelengths used to build the polynomials. If NULL (default), the column names of x are used when they are numeric, and the column indices otherwise.
robust: A logical value indicating whether the median (TRUE, default) or the mean spectrum is used as the reference.

Value

An object of class specproc_emsc (a list), which predict() applies to new spectra, with the following components:

correction: The corrected spectra.
coefficients: A tibble with the estimated coefficients of each spectrum: slope ($b_i$), offset ($a_i$), poly1, ... ($d_{ik}$) and interferent1, ... ($h_{ij}$).
reference: The reference spectrum.
design: The design matrix (one column per model term).

Details

Each spectrum $\textbf{x}_i$ is modeled as a linear combination of a reference spectrum $\textbf{m}$, polynomials of the wavelength $\lambda$ and interferent spectra $\textbf{k}_j$: $$\textbf{x}_i = b_i\textbf{m} + a_i + d_{i1}\lambda + d_{i2}\lambda^2 + \dots + \sum_j h_{ij}\textbf{k}_j + \textbf{e}_i$$ The coefficients are estimated by least squares, and the corrected spectrum is $$\textbf{x}_i^{corr} = (\textbf{x}_i - a_i - d_{i1}\lambda - \dots - \sum_j h_{ij}\textbf{k}_j) / b_i$$ With degree = 0 and no interferents, EMSC is identical to msc() with drop.offset = TRUE.

The wavelengths are rescaled to $[-1, 1]$ before the polynomials are formed, for numerical stability. Interferent spectra describe variation that should be removed, for example the spectrum of a contaminant or the dominant directions of repeated measurements of the same samples (see epo()).

The reference spectrum is estimated from x unless xref is given, so new spectra should be corrected with predict(), which uses the reference and design of the calibration data.

References

Martens, H., Stark, E. (1991). Extended multiplicative signal correction and spectral interference subtraction: new preprocessing methods for near infrared spectroscopy. Journal of Pharmaceutical and Biomedical Analysis, 9(8):625-635.
Afseth, N.K., Kohler, A. (2012). Extended multiplicative signal correction in vibrational spectroscopy, a tutorial. Chemometrics and Intelligent Laboratory Systems, 117:92-99.

Author

Christian L. Goueguel

Examples

data(forageLIBS)
spectra <- forageLIBS[-(1:14)]  # the spectral channels
fit <- emsc(spectra[1:300, ], degree = 2)
head(fit$coefficients)
#> # A tibble: 6 × 4
#>   slope offset  poly1  poly2
#>   <dbl>  <dbl>  <dbl>  <dbl>
#> 1 1.06   -89.9 104.    152. 
#> 2 1.02    35.2  88.9   -36.2
#> 3 1.05    58.3   9.46 -106. 
#> 4 1.13    41.2 -77.5  -310. 
#> 5 0.894   99.9  79.5    61.2
#> 6 1.000   54.8  32.3   -35.6
# new spectra are corrected with the calibration reference
corrected <- predict(fit, spectra[301:368, ])
dim(corrected)
#> [1]   68 7152