iProcrustes | R Documentation |
Based on generalized Procrustes analysis, this function determines a
linear transformation (rotation/reflection and scalling) of the points in
matrix x
to align them to their reference points
in matrix xbar
. The alignemnt is carried out by
minimizing the distance between the points in x
and xbar
.
iProcrustes(x, xbar, rotation.only=TRUE, scalling=TRUE, translate=FALSE)
x |
A numerical matrix to be align to points in |
xbar |
A numerical, reference matrix to which points in matrix |
rotation.only |
Logical. When |
scalling |
Logical. When |
translate |
Logical. Set |
Suppose the points in matrix X
and \bar{X}
are centralized
(meaning their centroids are at the origin). The
linear transformation of X
for aligning X
to its reference
matrix \bar{X}
., i.e., min ||sXQ - \bar{X}||_F
, is given by:
Q = VU^T,
and
s = trace(\bar{X}^TXQ) / trace(X^T X),
where V and U are the sigular value vectors of \bar{X}^T X
(that is,
\bar{X}^T X = U \Sigma V^T
), and s
is the scalling factor.
A list of the linear tranformation with items
Q |
An orthogonal, rotation/reflection matrix. |
scal |
A scalling factor |
.
T |
(optional) A translation vector used to shift the centroid of the
points in matrix |
T.xbar |
(optional) Centered |
Note that the return values of this function do not include the transformed
matrix scal* x* Q
or scal*(x-IT)*Q
, where T
is the
translation vector and I
is an n-by-1
vector with elements
1
.
C. J. Wong cwon2@fhcrc.org
gpaSet
## Example 1
x <- matrix(runif(20), nrow=10, ncol=2)+ 1.4
s <- matrix(c(cos(60), -sin(60), sin(60), cos(60)),
nrow=2, ncol=2, byrow=TRUE)
xbar <- 2.2 *(x %*% s) - 0.1
lt <- iProcrustes(x, xbar, translate=TRUE) ## return linear transformation
lt
## showing result
I <- matrix(1, nrow=nrow(x), ncol=1)
tx <- x - I %*% lt$T
## get the transformed matrix xnew
xnew <- lt$scal * (tx %*% lt$Q)
if (require(lattice)) {
xyplot(V1 ~ V2,
do.call(make.groups, lapply(list(x=x, xbar=xbar, T.xbar=lt$T.xbar,
xnew=xnew),as.data.frame)),
group=which, aspect=c(0.7), pch=c(1,3,2,4), col.symbol="black",
main=("Align the points in x to xbar"),
key=list(points=list(pch=c(1,3,2,4), col="black"), space="right",
text=list(c("x", "xbar", "T.xbar", "xnew"))))
}
## Example 2. centralized x and xbar prior to using iProcrustes
x <- matrix(runif(10), nrow=5, ncol=2)
s <- matrix(c(cos(60), -sin(60), sin(60), cos(60)),
nrow=2, ncol=2, byrow=TRUE)
xbar <- 1.2 *(x %*% s) - 2
I <- matrix(1, nrow=nrow(x), ncol=1)
x <- x-(I %*% colMeans(x)) ## shift the centroid of points in x to the origin
xbar <- xbar - (I %*% colMeans(xbar)) ## shift centroid to the origin
lt <- iProcrustes(x, xbar, translate=FALSE) ## return linear transformation
## only return the rotation/reflection matrix and scalling factor
lt
xnew=lt$scal *(x %*% lt$Q) ## transformed matrix aligned to centralized xbar
if (require(lattice)) {
xyplot(V1 ~ V2,
do.call(make.groups, lapply(list(x=x,xbar=xbar,
xnew=xnew), as.data.frame)),
group=which, auto.key=list(space="right"))
}
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