statsmodels.multivariate.factor_rotation.target_rotation¶

statsmodels.multivariate.factor_rotation.target_rotation(A, H, full_rank=False)[source]¶

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

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

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

Under the assumption that $A^{\ast }H$ has full rank, the analytical solution $T$ is given by:

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

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

$$A^{\ast }H=U\mathrm{\Sigma }V,$$

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