# 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:

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

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