supplementaryVariables4PLSC: Project supplementary variables (columns) for a PLSC analysis...

In HerveAbdi/data4PCCAR: Some data sets and R-functions for PCA and CA to accompany Abdi & Beaton (to appear, 2023) Principal Component and Correspondence Analysis with R

Project supplementary variables (columns) for a PLSC analysis (from tepPLS).

Description

supplementaryVariables4PLSC: Projects supplementary variables (columns) for a PLSC analysis (from tepPLS). The variables should be measured on the same observations as the observations measured on the original analysis. The original data consisted in 2 matrices denoted X (dimensions N by I) and Y (N by J). The supplementary data denoted Vsup is a N by K matrix, that can be considered as originating either from X (and then denoted Xsup) or Y (and then denoted Ysup) . If originating from X (resp, Y) matrix Y (resp, X) is the dual matrix. Note that only the dual matrix is needed to project supplementary variables. See details for more.

Usage

supplementaryVariables4PLSC(
  var.sup,
  resPLSC,
  Xset = NULL,
  Yset = NULL,
  X.center = TRUE,
  X.scale = "SS1",
  Y.center = TRUE,
  Y.scale = "SS1",
  dimNames = "Dimension "
)

Arguments

var.sup	Vsup: The N by K matrix of K supplementary variables.
resPLSC	the results of a PLSC analysis performed with tepPLS.
Xset	the original X (N by I) data matrix. If NULL, the supplementary data are projected on the dual set (i.e., Y). See also details for more.
Yset	the original Y (N by J) data matrix. If NULL, the supplementary data are projected on the dual set (i.e., X). See also details for more.
X.center	centering parameter for X (Default: TRUE). See expo.scale for details.
X.scale	scaling parameter for X (Default: 'SS1'). See expo.scale for details.
Y.center	centering parameter for Y (Default: TRUE). See expo.scale for details.
Y.scale	scaling parameter for Y (Default: 'SS1'). See expo.scale for details.
dimNames	Names for the dimensions (i.e., factors) for the supplementary loadings (Default: 'Dimension ').

Details

The computation relies on the SVD of the correlation matrix between X and Y, computed as R = X'Y (where X Y are the original data matrices that have been preprocessed, with, e.g., centering and scaling) and decomposed with the SVD as R = PDQ', with the usual constraints that P'P = Q'Q = I.

Active loadings

The active loadings are P (X-loadings) and Q (Y-loadings). These loadings come from the SVD of matrix R = X'Y = PDQ'.

Transition formulas

The loadings of one set can be obtained from the correlation matrix R and the loadings from the dual set. For example:

P = X'YQ inv(D) = RQ inv(D) (with inv(D) being the inverse of D). Eq. 1

Projection of supplementary variables

Supplementary variable loadings are obtained by first computing their correlation with their dual set and then projecting these on the singular vector of their dual set. So, for example, the loadings denoted Psup for an N by K matrix of K supplementary variables considered as belonging to the X-set will be projected by first computing the correlation matrix between these variables and Y (the dual set) as: Rsup = Xsup' Y (note that we assume here that Xsup has been pre-processed in the same way as X). The supplementary loadings are now computed by replacing in Eq.1 X by Xsup to obtain:

Psup = Xsup' * Y * Q * inv(D) = Rsup * Q * inv(D) . Eq.2.

Value

a list with the following elements:

• "loadings.sup.X": The loadings of the supplementary variables as originating from the Xset (needs to have the dual Yset to be computed).

• "sup.fi": The singular-value-normalized loadings of the supplementary variables as originating from the Xset (needs to have the dual Yset to be computed).

• "loadings.sup.Y": The loadings of the supplementary variables as originating from the Yset (needs to have the dual Xset to be computed).

• "sup.fj": The singular-value-normalized loadings of the supplementary variables as originating from the Yset (needs to have the dual Xset to be computed).

• "cor.lx": The correlations between the supplementary variables and the X set.

• "cor.ly": The correlations between the supplementary variables and the Y set.

Author(s)

Hervé Abdi

References

See:

Abdi, H., & Williams, L.J. (2013). Partial least squares methods: Partial least squares correlation and partial least square regression. In: B. Reisfeld & A. Mayeno (Eds.), Methods in Molecular Biology: Computational Toxicology. New York: Springer Verlag. pp. 549-579.

Abdi, H. (2007). Singular Value Decomposition (SVD) and Generalized Singular Value Decomposition (GSVD). In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage. pp. 907-912.

See Also

tepPLS expo.scale supplementaryObservations4PLSC

Examples

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if(interactive()){
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HerveAbdi/data4PCCAR documentation built on Sept. 11, 2022, 4:19 p.m.