statsmodels.multivariate.factor_rotation.target_rotation

statsmodels.multivariate.factor_rotation.target_rotation(A, H, full_rank=False)

Analytically performs orthogonal rotations towards a target matrix, i.e., we minimize:

$$\phi(L) =\frac{1}{2}\|AT-H\|^2.$$

where $T$ is an orthogonal matrix. This problem is also known as an orthogonal Procrustes problem.

Under the assumption that $A^*H$ has full rank, the analytical solution $T$ is given by:

$$T = (A^*HH^*A)^{-\frac{1}{2}}A^*H,$$

see Green (1952). In other cases the solution is given by $T = UV$, where $U$ and $V$ result from the singular value decomposition of $A^*H$:

$$A^*H = U\Sigma V,$$

see Schonemann (1966).

Parameters

Anumpy matrix (default None)

non rotated factors

Hnumpy matrix

target matrix

full_rankbool (default FAlse)

if set to true full rank is assumed

Returns

The matrix $T$.

References

[1] Green (1952, Psychometrika) - The orthogonal approximation of an oblique structure in factor analysis

[2] Schonemann (1966) - A generalized solution of the orthogonal procrustes problem

[3] Gower, Dijksterhuis (2004) - Procrustes problems