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External Parameter Orthogonalization

Source: R/epo.R
epo.Rd

This function implements the External Parameter Orthogonalization (EPO) algorithm as proposed by Roger et al. (2003). The EPO algorithm aims to remove interferences or clutter present in the spectral matrix, effectively separating the signal of interest from unwanted perturbations.

Usage

epo(x, ncomp = 2, clutter = NULL)

Arguments

x

A numeric matrix, data frame or tibble.

ncomp

An integer specifying the number of singular vectors to orthogonalize against. Default is 2.

clutter

An optional numeric matrix or data frame, with the same number of columns as x, describing the external (clutter) variation. If NULL (default), the clutter directions are estimated from x itself, i.e. the ncomp dominant directions of x are removed.

Value

An object of class specproc_epo (a list), which predict() applies to new spectra, with four components:

  • correction: The orthogonalized matrix, representing the signal of interest.

  • clutter: The clutter part of x, \(\textbf{XVV}^T\).

  • loadings: The singular vectors \(\textbf{V}\) used for the orthogonalization.

  • singular_values: The corresponding singular values of the clutter matrix.

Details

The EPO algorithm works by detecting the dominant directions (right singular vectors) of a clutter matrix \(\textbf{D}\) that describe the external variations, e.g. the differences between spectra of the same samples measured under different temperatures or moisture levels. It then projects the spectral matrix onto the subspace orthogonal to these variations, effectively removing the unwanted perturbations and extracting the signal of interest.

Let \(\textbf{X}\) be the spectral matrix. The EPO algorithm aims to split \(\textbf{X}\) into:

$$\textbf{X} = \textbf{XP} + \textbf{XQ} + \textbf{R}$$

where \(\textbf{P}\) and \(\textbf{Q}\) are, respectively, the projection matrices of \(\textbf{X}\) onto the useful and perturbation (clutter) subspaces. \(\textbf{R}\) is the residual matrix. With \(\textbf{V}\) the first ncomp right singular vectors of \(\textbf{D}\), \(\textbf{Q} = \textbf{VV}^T\) and the corrected matrix is \(\textbf{X} - \textbf{XVV}^T\).

The singular value decomposition is computed in C++ (Eigen), and the \(p \times p\) projection matrix is never formed explicitly.

References

  • Roger, J.-M., Chauchard, F., Bellon-Maurel, V. (2003). EPO-PLS external parameter orthogonalization of PLS application to temperature-independent measurement of sugar content of intact fruits. Chemometrics and Intelligent Laboratory Systems, 66(2):191-204.

See also

predict.specproc_filter() to correct new spectra with the fitted filter, and step_epo() to use it in a tidymodels recipe.

Author

Christian L. Goueguel

Examples

data(forageLIBS)
spectra <- forageLIBS[-(1:14)]  # the spectral channels
# the three samples measured twice
twice <- forageLIBS$Sample[duplicated(forageLIBS$Sample)]
first <- match(twice, forageLIBS$Sample)
second <- vapply(twice, function(s) max(which(forageLIBS$Sample == s)), integer(1))
# their differences describe the variation between repeated measurements
clutter <- as.matrix(spectra[second, ]) - as.matrix(spectra[first, ])
res <- epo(spectra, ncomp = 1, clutter = clutter)
dim(res$correction)
#> [1]  368 7152