supplementaryObservations4PLSCA: compute the latent variables for supplementary observations...

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

compute the latent variables for supplementary observations for a PLSCA model computed with tepPLSCA.

Description

supplementaryObservations4PLSCA: computes the latent variables for supplementary observations for a PLSCA model computed with tepPLSCA.

Usage

supplementaryObservations4PLSCA(
  resPLSCA,
  Xsup = NULL,
  Ysup = NULL,
  dimNames = "Dimension "
)

Arguments

resPLSCA	the results of a PLSCA analysis from tepPLSCA.
Xsup	an Nsup by I matrix of supplementary observations matching the X matrix (see description for details). When NULL (Default) nothing is computed for Xsup.
Ysup	an Nsup by J matrix of supplementary observations matching the Y matrix (see description for details). When NULL (Default) nothing is computed for Ysup.
dimNames	Names for the dimensions (i.e., factors) for the supplementary loadings (Default: 'Dimension ').

Details

The original analysis is performed with tepPLSCA on the original data matrices X (N by I) and Y (N by J). The supplementary data matrices should have I columns for Xsup and J columns for Ysup. Note that PLSCA is used with qualitative variables (i.e., factors) recoded as 0/1 variables with disjunctive coding (i.e., with makeNominalData), the supplementary variables need to be recoded in the same way.

Implementation

For PLSCA the observations need to be pre-processed in the same way as the original observations. Often, in PLSCA, the observations are described by qualitative variables (in general coded as factors) which are then recoded (e.g., with the function makeNominalData from ExPosition) as a set of 0/1 vectors prior to ruccing PLSCA. So When this , the supplementary observations should becoded as factors too with the same levels (aka modalities) as the active observations (see also makeNominalData).

Computation

The projections of supplementary observations in PLSC is obtained using the standard transition formulas from correspondence analysis (with an additional scaling factor to get the covariance of the latent variables equal to their singular values).

Transition formulas

The latent variables can be obtained from the loadings of their set. For example: if we denote Delta the diagonal matrix of the singular values, F (resp. G) the singular value normalized loadings (denoted fi, resp. fj, in PLSCA), and Lx (resp. Ly) the row (resp. column) latent variables (called lx and ly in tepLSCA), the latent variables of one set are derived from the set loadings:

Lx = sqrt(N) XF inv(Delta) and Ly = sqrt(N) YG inv(Delta). Eq.1

with: inv(Delta) being the inverse of Delta, N being the number of rows (i.e., observations) of X and Y, and X and Y are row profile versions of the original data sets.

Projection of supplementary observations

Supplementary observations latent variable values are obtained by using the transition formulas from correspondence analysis (see Eq.1, Section above). So, the values for the latent variable for the supplementary observations from the Xset and the Yset can be obtained from their row profiles (denoted Xsup and Ysup) by replacing in Eq.1 X by Xsup and Y by Ysup:

Lxsup = sqrt(N) Xsup F inv(Delta) and Lysup = sqrt(N) Ysup G inv(Delta). Eq.2

Value

A list with lx.sup and ly.sup giving the latent variables values of the supplementary observations for (respectively) X and Y.

Author(s)

Hervé Abdi

References

See:

Beaton, D., Dunlop, J., ADNI, & Abdi, H. (2016). Partial Least Squares-Correspondence Analysis (PLSCA): A framework to simultaneously analyze behavioral and genetic data. Psychological Methods, 21, 621-651.

Abdi H. & Béra, M. (2018). Correspondence analysis. In R. Alhajj and J. Rokne (Eds.), Encyclopedia of Social Networks and Mining (2nd Edition). New York: Springer Verlag.

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

tepPLSCA makeRowProfiles supplementaryVariables4PLSCA supplementaryObservations4PLSC

HerveAbdi/data4PCCAR documentation built on Sept. 11, 2022, 4:19 p.m.