Random.Start {GPArotation}

Generate a Random Orthogonal Rotation

Description

Random orthogonal rotation to use as Tmat matrix to start GPFRSorth, GPFRSoblq, GPForth, or GPFoblq.

Usage

    Random.Start(k)
    

Arguments

k

An integer indicating the dimension of the square matrix.

Details

The random start function produces an orthogonal matrix with columns of length one based on the QR decompostion. This randomization procedures follows the logic of Stewart(1980) and Mezzari(2007), as of GPArotation version 2024.2-1.

Value

An orthogonal matrix.

Author(s)

Coen A. Bernaards and Robert I. Jennrich with some R modifications by Paul Gilbert. Additional input from Yves Rosseel.

References

Stewart, G. W. (1980). The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators. SIAM Journal on Numerical Analysis, 17(3), 403–409. http://www.jstor.org/stable/2156882

Mezzadri, F. (2007). How to generate random matrices from the classical compact groups. Notices of the American Mathematical Society, 54(5), 592–604. https://arxiv.org/abs/math-ph/0609050

See Also

GPFRSorth, GPFRSoblq, GPForth, GPFoblq, rotations

Examples

  # Generate a random ortogonal matrix of dimension 5 x 5
  Random.Start(5)
  
  # function for generating orthogonal or oblique random matrix
  Random.Start <- function(k = 2L,orthogonal=TRUE){
    mat <- matrix(rnorm(k*k),k)
    if (orthogonal){
      qr.out <- qr(matrix(rnorm(k * k), nrow = k, ncol = k))
      Q <- qr.Q(qr.out)
      R <- qr.R(qr.out)
      R.diag <- diag(R)
      R.diag2 <- R.diag/abs(R.diag)
      ans <- t(t(Q) * R.diag2)
      ans
      }
    else{
	  ans <- mat %*% diag(1/sqrt(diag(crossprod(mat))))
 	  }
    ans
    }
    	
  data("Thurstone", package="GPArotation")
  simplimax(box26,Tmat = Random.Start(3, orthogonal = TRUE))
  simplimax(box26,Tmat = Random.Start(3, orthogonal = FALSE))

  # covariance matrix is Phi = t(Th) %*% Th
  rms <- Random.Start(3, FALSE)
  t(rms) %*% rms # covariance matrix because oblique rms
  rms <- Random.Start(3, TRUE)
  t(rms) %*% rms # identity matrix because orthogonal rms