GeometricMorphometricsMix (version 0.6.0.1)

ProjectOrthogonal: Project to subspace orthogonal to a vector

Description

This function projects data to the subspace orthogonal to a multivariate column vector

Usage

ProjectOrthogonal(Data, vector)

Value

The function outputs a matrix n x p of the original data projected to the subspace orthogonal to the vector

Arguments

Data

matrix n x p of n observation for p variables,

vector

column vector (matrix p x 1 of p variables)

Details

This function is useful to remove from a dataset the variation along a specific direction (e.g., a principal component). It has been used extensively for many applications, such as

• 'Size Correction' (removal of an allometric vector), also called 'Burnaby's method (Burnaby 1966; see also Rohlf & Bookstein 1987)

• Remove body arching from fish (Valentin et al. 2008; see also Fruciano 2016 for a discussion and other examples of usage in the context of measurement error in geometric morphometrics)

• Removing variation due to sexual dimorphism on a set of individuals with unknown sex (Fruciano et al. 2014)

Optionally, vector can also be a matrix with more than one column (in this case the Data is projected to the subspace orthogonal to the space spanned by all dimensions in vector)

References

Burnaby T. 1966. Growth-invariant discriminant functions and generalized distances. Biometrics:96-110.

Fruciano C. 2016. Measurement error in geometric morphometrics. Development Genes and Evolution 226:139-158.

Fruciano C, Pappalardo AM, Tigano C, Ferrito V. 2014. Phylogeographical relationships of Sicilian brown trout and the effects of genetic introgression on morphospace occupation. Biological Journal of the Linnean Society 112:387-398.

Rohlf FJ, Bookstein FL. 1987. A Comment on Shearing as a Method for' Size Correction'. Systematic Zoology:356-367.

Valentin AE, Penin X, Chanut JP, Sévigny JM, Rohlf FJ. 2008. Arching effect on fish body shape in geometric morphometric studies. Journal of Fish Biology 73:623-638.

Examples

library(MASS)
A=mvrnorm(50,mu=rep(1,50),Sigma=diag(50))
B=mvrnorm(50,mu=rep(0,50),Sigma=diag(50))
AB=rbind(A,B)
Group=as.factor(c(rep(1,50),rep(2,50)))
# Create two groups of observations (e.g., specimens)
# one centered at 0 and the other at 1
# and combine them in a single sample

PCA=prcomp(AB)
# Combine the two groups and perform a PCA

plot(PCA$x[,1],PCA$x[,2], asp=1, col=Group)
# Plot the scores along the first two principal components
# The two groups are clearly distinct (red and black)

ABproj=ProjectOrthogonal(AB,cbind(PCA$rotation[,1]))
# Project the original data (both groups)
# to the subspace orthogonal to the first principal component
# (which is the direction along which there is most of variation
# among groups)

PCAproj=prcomp(ABproj)
# Perform a new PCA on the 'corrected' dataset

plot(PCAproj$x[,1], PCAproj$x[,2], asp=1, col=Group)
# Plot the scores along the first two principal components
# of the 'corrected' data
# Notice how the two groups are now pretty much
# indistinguishable