random initial values for rotation matrix in GPA rotations

factor_analyzer/rotator.py の Rotator._oblique、Rotator._orthogonal、Rotator._varimax から始め、他の回転メソッドについても同じ初期化になっているか確認してください。現在の単位行列の挙動を検証し、random_state が直交初期化をどのように制御するかを定義し、デフォルトが変更されないことを確認してください。

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)