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R: Procrustes analysis. Using singular value decomposition (SVD)...

iProcrustes

R Documentation

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`.

Details

Suppose the points in matrix X and ar{X} are centralized (meaning their centroids are at the origin). The linear transformation of X for aligning X to its reference matrix ar{X}., i.e., min ||sXQ - ar{X}||_F, is given by:

Q = VU^T,

and

s = trace(ar{X}^TXQ) / trace(X^T X),

where V and U are the sigular value vectors of ar{X}^T X (that is, ar{X}^T X = U Σ V^T), and s is the scalling factor.

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.

Author(s)

C. J. Wong cwon2@fhcrc.org

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"))
}

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(flowStats)
Loading required package: flowCore
Loading required package: fda
Loading required package: splines
Loading required package: Matrix

Attaching package: 'Matrix'

The following object is masked from 'package:flowCore':

    %&%


Attaching package: 'fda'

The following object is masked from 'package:graphics':

    matplot

Loading required package: mvoutlier
Loading required package: sgeostat
sROC 0.1-2 loaded
Loading required package: cluster
Loading required package: flowWorkspace
Loading required package: flowViz
Loading required package: lattice
Loading required package: ncdfFlow
Loading required package: RcppArmadillo
Loading required package: BH
Loading required package: gridExtra
> png(filename="/home/ddbj/snapshot/RGM3/R_BC/result/flowStats/iProcrustes.Rd_%03d_medium.png", width=480, height=480)
> ### Name: iProcrustes
> ### Title: 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.
> ### Aliases: iProcrustes
> 
> ### ** 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
$T
[1] 1.881520 1.897843

$Q
           [,1]       [,2]
[1,] -0.9524130  0.3048106
[2,] -0.3048106 -0.9524130

$scal
[1] 2.2

$T.xbar
            [,1]        [,2]
 [1,]  1.0151945  0.52036693
 [2,]  0.1876105 -0.97380402
 [3,] -1.3480336 -0.47899586
 [4,]  0.9808552 -0.07904441
 [5,]  0.5582530 -0.11413573
 [6,] -0.8546179  0.31452015
 [7,] -0.3861145 -0.31097066
 [8,] -0.6135420  0.44992692
 [9,] -0.5400927  0.57093051
[10,]  1.0004874  0.10120617

> 
> ## 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
$Q
           [,1]       [,2]
[1,] -0.9524130  0.3048106
[2,] -0.3048106 -0.9524130

$scal
[1] 1.2

> 
> 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"))
+ }
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>