y-Gradient Generalized Least Squares Weighting

The y-gradient generalized least squares weighting algorithm (GLSW) removes variance from the data (spectra), which is orthogonal to the response.

Usage

y_gradient_glsw(x, y, alpha = 0.01, window = 5)

Arguments

x: A numeric matrix, data frame or tibble, representing the predictors data.
y: A numeric vector representing the response vector.
alpha: A positive numeric value specifying the weighting parameter. Typical values range from 1 to 0.0001. Default is 0.01.
window: An odd integer giving the width of the Savitzky-Golay window used to compute the gradients. Default is 5.

Value

A tibble containing the \(p \times p\) filtering matrix.

Details

The y-Gradient GLSW is an alternative method to GLSW, where a continuous \(\textbf{y}\)-variable is used to develop pseudo-groupings of samples in \(\textbf{X}\) by comparing the differences in the corresponding \(\textbf{y}\) values. This is referred to as the "gradient method" because it utilizes a gradient of the sorted \(\textbf{X}\)- and \(\textbf{y}\)-blocks to calculate a covariance matrix.

The samples are sorted by increasing \(\textbf{y}\), and the first derivatives of \(\textbf{X}\) and \(\textbf{y}\) along the sample axis are computed with a Savitzky-Golay filter. Samples whose neighbors have similar \(\textbf{y}\) values receive large weights \(w_i = 2^{-\Delta y_i / s_{\Delta y}}\), so the differences between their spectra, \(\Delta\textbf{X}\), describe variation unrelated to \(\textbf{y}\). The filter is then built as in glsw() from \(\textbf{C} = \Delta\textbf{X}^T\textbf{W}^2\Delta\textbf{X}\).

References

Zorzetti, B.M., Shaver, J.M., Harynuk, J.J., (2011). Estimation of the age of a weathered mixture of volatile organic compounds. Analytica Chimica Acta, 694(1-2):31–37.

Author

Christian L. Goueguel

Examples

data(forageLIBS)
spectra <- forageLIBS[-(1:14)]
wl <- as.numeric(names(spectra))
x <- spectra[wl > 760 & wl < 780]  # the K I resonance lines
G <- y_gradient_glsw(x, forageLIBS$K, alpha = 0.01)
filtered <- as.matrix(x) %*% as.matrix(G)
dim(filtered)
#> [1] 368 245