Random.Start: Random Starting Matrix for Factor Rotation

In GPArotation: Gradient Projection Factor Rotation

Random Starting Matrix for Factor Rotation

Description

Generates a random orthogonal matrix for use as an initial rotation matrix (Tmat) in gradient projection rotation functions.

Usage

    Random.Start(k = 2L)

Arguments

k	a positive integer specifying the dimension of the square orthogonal matrix to generate (default 2).

Details

The function generates a random orthogonal matrix using QR decomposition of a matrix of standard normal variates, with a sign correction applied to the diagonal of R to ensure uniform sampling from the Haar measure. This follows the approach of Stewart (1980) and Mezzadri (2007).

The naive approach of qr.Q(qr(matrix(rnorm(k*k), k))) does not guarantee uniform sampling from the Haar measure. The sign correction Q %*% diag(sign(diag(R))) is required to achieve this. This was updated in GPArotation 2024.2-1 following a suggestion by Yves Rosseel.

For oblique rotation a random starting transformation matrix can be generated by normalizing the columns of a random matrix: X %*% diag(1/sqrt(diag(crossprod(X)))) where X <- matrix(rnorm(k*k), k).

Value

A k \times k orthogonal matrix drawn uniformly from the Haar measure on the orthogonal group O(k). Columns have unit length and are mutually orthogonal.

Author(s)

Coen A. Bernaards and Robert I. Jennrich with some R modifications by Paul Gilbert. Updated following a suggestion by 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. doi: 10.1137/0717034

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

See Also

GPFRSorth, GPFRSoblq, GPForth, GPFoblq, rotations

Examples

  # Generate a 5 x 5 random orthogonal matrix
  Random.Start(5)

  # Verify orthogonality: t(Q) %*% Q should be the identity matrix
  Q <- Random.Start(4)
  round(t(Q) %*% Q, 10)

  # Use as starting matrix for rotation
  data("Thurstone", package = "GPArotation")
  simplimax(box26, Tmat = Random.Start(3))

GPArotation documentation built on April 29, 2026, 9:08 a.m.