flowStats (version 3.30.0)

iProcrustes: Procrustes analysis. Using singular value decomposition (SVD) to determine a linear transformation to align the points in X to the points in a reference matrix Y.

Description

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.

Usage

iProcrustes(x, xbar, rotation.only=TRUE, scalling=TRUE, translate=FALSE)

Arguments

x

A numerical matrix to be align to points in xbar, the second arguement. The columns represents the coordinates of the points. The matrices x and xbar must have the same dimensions.

xbar

A numerical, reference matrix to which points in matrix x are to be aligned.

rotation.only

Logical. When rotaion.only is TRUE, it allows the function to lose reflection component of the linear transformation. Although it might not give the best-fitting aligenment, when dealing with flow cytometry data alignment, a non-reflection transformation is prefered. When rotaion.only is FALSE, it allows the function to retain the reflection component.

scalling

Logical. When scalling is FALSE, it allows the function to calculate the linear transformation without a scalling factor. That is, the returning scalling factor is set to $1$.

translate

Logical. Set translate to FALSE when the points in matrices x and xbar are already centralized prior to applying this function. When translate is TRUE, it allows the function to translate the centroid the points in matrix x to that of points in xbar.

Value

• 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 x to the origin. Returned when translate is TRUE.
• T.xbar
  (optional) Centered xbar (that is, the centroid of the points in xbar is translated to the origin). Returned when translate is TRUE.
• 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 $1$.

Details

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.

See Also

gpaSet

Examples

## 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"))
}