random initial values for rotation matrix in GPA rotations

[szorowi1]

Hi all. First, thanks so much to the devs of factor_analyzer. It's a fantastic and much needed package!

Is your feature request related to a problem? Please describe.

As has been pointed out, the GPA rotation methods often converge to local minima (Browne, 2001; Bernaards & Jennrich, 2005; Nguyen & Waller, 2022). As such, it's been recommended researchers initialize the GPA from many starting values when searching for a global minimum. I'm wondering if the Rotator class could be amended in order to allow initializing the GPA from different starting values.

Describe the solution you'd like

One simple solution would be for the Rotator class to accept a new argument, random_state, which defaults to None. When random_state = None, the rotation matrix is initialized with np.eye(n_cols) as is currently implemented (L326, L416, L493). When a user specifies a seed, however, the rotation matrix is initialized as a random orthogonal matrix from scipy.stats.ortho_group. That is,:

# initialize the rotation matrix
_, n_cols = loadings.shape
if self.random_state == None:
    rotation_matrix = np.eye(n_cols)
else:
    rotation_matrix = sp.stats.ortho_group(dim=n_cols, seed=self.random_state).rvs()

A solution like the above could be added to the Rotator._oblique, Rotator._orthogonal, Rotator._varimax methods (and any others that I missed).

Apologies in advance if the above is already possible and I've simply missed how. If the above sounds like it may be useful, I'd be happy to attempt a PR.

All the best,
Sam (@szorowi1)

[desilinguist]

Hi @szorowi1, this is a great idea! We will try to include this in the next release.

[szorowi1]

Awesome, I'm happy to hear you think it's a good idea! I opened a PR (#117) to get this started. I've modified the Rotator class as suggested above.