pcar
Is there any possibility of calculating PCA rotation similar to R's prcomp function, by that I mean if I do the following
log.ir <- log(iris[, 1:4])
ir.species <- iris[, 5]
ir.pca <- prcomp(log.ir)$x
where ir.pca is a 150 by 4 (row by column) matrix, can I use this information to calculate rotation without using prcomp(log.ir)$rotation?
In a 2D case, the rotation matrix is $2\times 2$ and contains two eigenvectors as its columns. The first eigenvector $(x,y)$ is given by the first column. Its angle to the horizontal axis (abscissa) is given by $$\alpha = \operatorname{arctan} (y/x).$$As all eigenvectors are scaled to have unit length, this is equal to $$\alpha = \operatorname{arccos} (x/1)=\operatorname{arccos} (x).$$
You might call it a "counterclockwise" angle "from the East direction". If you want a "clockwise" angle "from the North direction", then you need $$\beta = 90^\circ - \alpha=90^\circ - \operatorname{arccos} (x)=
\operatorname{arcsin} (x).$$
In R, it should be:
beta = asin(pc$rotation[1,1])*180/pi
which is more or less your formula too.
I don't know exactly what robCompositions is doing. Remember, that this package is applying robust statistics.
If you want, you can calculate the first principal coordinate using the coda.base package, or calculate this orthornormal coordinates from the clr-coordinates.
coda.base# I am using the first three variables of iris dataset as a composition
X = iris[,1:3]
library(coda.base)
H_pc = coordinates(X, 'pc')
# You have the first coordinate in
H_pc[,1]
The basis is obtained directly from the principal coordinate basis.
attr(H_pc, 'basis')
## Which is equivalent to
H_clr = coordinates(X, 'clr')
prcomp(H_clr)
To obtain the first principal component you can use the following code
lX = log(X)
clrX = lX - rowMeans(lX)
B = prcomp(clrX)
as.matrix(clrX) %*% B$rotation[,1]
Best Answer
The principal components of your $N\times p$ (centered) data matrix $\mathbf{X}$ are given by $$ \mathbf{Z} = \mathbf{X V} $$ So you have that $$ \mathbf{V}' = \mathbf Z'\mathbf X (\mathbf X'\mathbf X) ^ {-1}. $$ You can compare the output of the second and third command: