kcomp: K-component analysis

In RRPP: Linear Model Evaluation with Randomized Residuals in a Permutation Procedure

K-component analysis

Description

Function performs a K-comonent analysis, which is an eigen decomposition of a ratio of ordinary least-squares (OLS) to generalized least squares (GLS) covariance matrices.

Usage

kcomp(Y, Cov, transform. = TRUE, scale. = FALSE, tol = NULL, rank. = NULL)

Arguments

Y	An n x p data matrix.
Cov	An required n x n covariance matrix to describe the non-independence among observations in Y, and provide a GLS-centering of data.
transform.	An optional argument to transform GLS-centered residuals, if TRUE. If FALSE, only GLS-centering is performed.
scale.	A logical value indicating whether the variables should be scaled to have unit variance before the analysis takes place. The default is FALSE.
tol	A value indicating the magnitude below which components should be omitted. (Components are omitted if their standard deviations are less than or equal to tol times the standard deviation of the first component.)
rank.	Optionally, a number specifying the maximal rank, i.e., maximal number of K components to be used. This argument can be set as alternative or in addition to tol, useful notably when the desired rank is considerably smaller than the dimensions of the matrix of the K matrix.

Details

The function performs a K-component analysis (KCA), as described in Mitteroecker et al. (2025). The analysis is similar to many that can be performed with the ordinate function, but whereas that function finds ordinate scores for a rotation or shear of a data space, KCA performs a relative eigen decomposition of a matrix product that represents a ratio between two covariance matrices (Mitteroecker and Bookstein, 2014). The eigenvalues are helpful for understanding the importance of the GLS estimation of a covariance matrix. For example, if the GLS estimation is made with respect to a matrix of phylogenetic covariances, the eigenvalues express the distribution of phylogenetic signal across components of the data space.

The matrix product (K matrix) is the inverse of the OLS covariance matrix times the GLS covariance matrix. Redundancies between these matrices will mean the K matrix is likely not full rank. Therefore, an algorithm is used to compute scores in appropriate dimensions. First, data are aligned to the square-root matrix of the GLS covariance matrix (Collyer and Adams, 2021), using the ordinate function. Second, relative eigen decomposition is performed, and the number of real eigenvectors that can be computed are retained. Lastly, the same number of aligned components are used for projection of data onto relative eigenvectors (the K components).

Value

An object of class kcomp is a list containing the following

values	The eigenvalues of the K matrix,
vectors	The eigenvectors of the K matrix.
scores	The projected scores of data onto the eigenvectors.

Author(s)

Michael Collyer

References

Collyer, M.L. and D.C. Adams. 2021. Phylogenetically-aligned Component Analysis. Methods in Ecology and evolution. In press.

Bookstein, F. L., & Mitteroecker, P. (2014). Comparing covariance matrices by relative eigenanalysis, with applications to organismal biology. Evolutionary biology, 41, 336-350.

Mitteroecker, P., Collyer, M. L., & Adams, D. C. (2024). Exploring Phylogenetic Signal in Multivariate Phenotypes by Maximizing Blomberg’s K. Systematic Biology, syae035.

See Also

plot.kcomp, ordinate, plot.default, rrpp.data.frame

Examples

 data(PlethMorph)
 PCA <- ordinate(as.data.frame(PlethMorph[c("TailLength",
 "HeadLength", "Snout.eye", "BodyWidth","Forelimb", "Hindlimb")],
 scale. = TRUE))
 Y <- PCA$x
 KCA <- kcomp(Y, Cov = PlethMorph$PhyCov, tol = 0.001)
 

RRPP documentation built on April 4, 2025, 3:45 a.m.