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Modified Orthogonal Projections to Latent Structures

Source: R/o2pls.R
o2pls.Rd

This function implements the modified orthogonal projections to latent structures (O2PLS) algorithm as proposed by Trygg (2002). The OPLS and O2PLS methods differ as follows: OPLS is unidirectional \((X \Rightarrow Y)\), meaning that only orthogonal variations in the \(X\)-space are filtered out. Whilst O2PLS is bi-directional \((X \Leftrightarrow Y)\), meaning that orthogonal variations in both the \(X\)- and \(Y\)-space are filtered out.

Usage

o2pls(x, y, ncomp = 1, nx = 1, ny = 0, center = TRUE, scale = FALSE)

Arguments

x

A numeric matrix or data frame representing the predictor variables.

y

A numeric vector, matrix or data frame representing the response variables.

ncomp

An integer giving the number of joint (predictive) components. Default is 1.

nx

An integer giving the number of \(Y\)-orthogonal components removed from x. Default is 1.

ny

An integer giving the number of \(X\)-orthogonal components removed from y. Default is 0.

center

A logical value indicating whether to mean-centered x and y. Default is TRUE.

scale

A logical value indicating whether to scale x and y. Default is FALSE.

Value

An object of class o2pls, a list containing:

  • correction: The filtered x (\(Y\)-orthogonal variation removed).

  • correction_y: The filtered y (\(X\)-orthogonal variation removed).

  • scores: A list with the joint scores x (\(\textbf{T}\)) and y (\(\textbf{U}\)), and the orthogonal scores x_ortho and y_ortho.

  • loadings: A list with the joint loadings x and y, and the orthogonal loadings x_ortho and y_ortho.

  • weights: A list with the joint weights x (\(\textbf{W}\)) and y (\(\textbf{C}\)), and the orthogonal weights x_ortho and y_ortho.

  • center, scale: The column centers and scales applied to x.

Details

The O2PLS method handles situations where systematic \(X\)-orthogonal variation in \(Y\) exists, and it is predictive in both ways, \((X \Rightarrow Y)\) and \((Y \Rightarrow X)\). The systematic part in \(X\) and \(Y\) is divided into two parts, one which is related to both \(X\) and \(Y\) (joint/covarying) and one that is not (orthogonal).

The algorithm follows Trygg and Wold (2003):

  1. The joint weights \(\textbf{W}\) and \(\textbf{C}\) are the dominant singular vectors of \(\textbf{Y}^T\textbf{X}\).

  2. For each \(Y\)-orthogonal component of \(X\), the weight \(\textbf{w}_{o}\) is the dominant singular vector of \(\textbf{E}_{XY}^T\textbf{T}\) (with \(\textbf{T} = \textbf{XW}\) and \(\textbf{E}_{XY} = \textbf{X} - \textbf{TW}^T\)), and \(\textbf{X}\) is deflated by \(\textbf{t}_o\textbf{p}_o^T\).

  3. The same procedure is applied to \(Y\) for the \(X\)-orthogonal components.

  4. The joint components are recomputed from the filtered data.

For a single response variable, ny must be 0 and the method is equivalent to OPLS with nx orthogonal components.

References

  • Trygg, J., (2002). O2-PLS for qualitative and quantitative analysis in multivariate calibration. J. Chemom. 16(1):283–293.

  • Trygg, J., Wold, S., (2003). O2-PLS, a two-block (X–Y) latent variable regression (LVR) method with an integral OSC filter. J. Chemom. 17(1):53–64.

See also

predict.specproc_filter() to correct new spectra with the fitted filter, and step_o2pls() to use it in a tidymodels recipe, with one or several outcomes.

Author

Christian L. Goueguel

Examples

data(forageLIBS)
spectra <- forageLIBS[-(1:14)]  # the spectral channels
# potassium and calcium at once
fit <- o2pls(spectra, forageLIBS[c("K", "Ca")], ncomp = 2, nx = 2)
fit
#> An object of class 'o2pls'
#> 
#> Joint components:        2
#> X-orthogonal components: 2
#> Y-orthogonal components: 0
#> 
#> - correction:  368 x 7152
#> - X weights:   7152 x 2
#> - Y weights:   2 x 2
#> - X scores:    368 x 2
#> - X ortho:     368 x 2