statsmodels.multivariate.factor_rotation.rotate_factors#

statsmodels.multivariate.factor_rotation.rotate_factors(A, method, *method_args, **algorithm_kwargs)[source]#

Subroutine for orthogonal and oblique rotation of the matrix \(A\)

For orthogonal rotations \(A\) is rotated to \(L\) according to

\[L = AT,\]

where \(T\) is an orthogonal matrix. And, for oblique rotations \(A\) is rotated to \(L\) according to

\[L = A(T^*)^{-1},\]

where \(T\) is a normal matrix.

Parameters:
Andarray

non rotated factors

methodstr

should be one of the methods listed below

method_argstuple

additional arguments that should be provided with each method

algorithm_kwargsdict
algorithm{‘gpa’, ‘gpa_der_free’, ‘analytic’}, optional

should be one of:

  • ‘gpa’: a numerical method

  • ‘gpa_der_free’: a derivative free numerical method

  • ‘analytic’ : an analytic method

Depending on the algorithm, there are algorithm specific keyword arguments. For the gpa and gpa_der_free, the following keyword arguments are available:

max_triesint, optional

maximum number of iterations. The default is 501.

tolfloat, optional

stop criterion, algorithm stops if Frobenius norm of gradient is smaller then tol

For analytic, the supported arguments depend on the method, see above.

See the lower level functions for more details.

Returns:
Lndarray

The rotated factors.

Tndarray

The rotation matrix.

Notes

What follows is a list of available methods. Depending on the method additional argument are required and different algorithms are available. The algorithm_kwargs are additional keyword arguments passed to the selected algorithm (see the parameters section). Unless stated otherwise, only the gpa and gpa_der_free algorithm are available.

Below,

  • \(L\) is a \(p\times k\) matrix;

  • \(N\) is \(k\times k\) matrix with zeros on the diagonal and ones elsewhere;

  • \(M\) is \(p\times p\) matrix with zeros on the diagonal and ones elsewhere;

  • \(C\) is a \(p\times p\) matrix with elements equal to \(1/p\);

  • \((X,Y)=\operatorname{Tr}(X^*Y)\) is the Frobenius norm;

  • \(\circ\) is the element-wise product or Hadamard product.

obliminorthogonal or oblique rotation that minimizes
\[\phi(L) = \frac{1}{4}(L\circ L,(I-\gamma C)(L\circ L)N).\]

For orthogonal rotations:

  • \(\gamma=0\) corresponds to quartimax,

  • \(\gamma=\frac{1}{2}\) corresponds to biquartimax,

  • \(\gamma=1\) corresponds to varimax,

  • \(\gamma=\frac{1}{p}\) corresponds to equamax.

For oblique rotations:

  • \(\gamma=0\) corresponds to quartimin,

  • \(\gamma=\frac{1}{2}\) corresponds to biquartimin.

method_args:

gammafloat

oblimin family parameter

rotation_method{‘orthogonal’, ‘oblique’}

should be one of {orthogonal, oblique}

orthomax : orthogonal rotation that minimizes

\[\phi(L) = -\frac{1}{4}(L\circ L,(I-\gamma C)(L\circ L)),\]

where \(0\leq\gamma\leq1\). The orthomax family is equivalent to the oblimin family (when restricted to orthogonal rotations). Furthermore,

  • \(\gamma=0\) corresponds to quartimax,

  • \(\gamma=\frac{1}{2}\) corresponds to biquartimax,

  • \(\gamma=1\) corresponds to varimax,

  • \(\gamma=\frac{1}{p}\) corresponds to equamax.

method_args:

gammafloat

orthomax family parameter, between 0 and 1

CF : Crawford-Ferguson family for orthogonal and oblique rotation which minimizes:

\[\phi(L) =\frac{1-\kappa}{4} (L\circ L,(L\circ L)N) -\frac{1}{4}(L\circ L,M(L\circ L)),\]

where \(0\leq\kappa\leq1\). For orthogonal rotations the oblimin (and orthomax) family of rotations is equivalent to the Crawford-Ferguson family. To be more precise:

  • \(\kappa=0\) corresponds to quartimax,

  • \(\kappa=\frac{1}{p}\) corresponds to varimax,

  • \(\kappa=\frac{k-1}{p+k-2}\) corresponds to parsimax,

  • \(\kappa=1\) corresponds to factor parsimony.

method_args:

kappafloat

Crawford-Ferguson family parameter, between 0 and 1

rotation_method{‘orthogonal’, ‘oblique’}

should be one of {orthogonal, oblique}

quartimaxorthogonal rotation method

minimizes the orthomax objective with \(\gamma=0\)

biquartimaxorthogonal rotation method

minimizes the orthomax objective with \(\gamma=\frac{1}{2}\)

varimaxorthogonal rotation method

minimizes the orthomax objective with \(\gamma=1\)

equamaxorthogonal rotation method

minimizes the orthomax objective with \(\gamma=\frac{1}{p}\)

parsimaxorthogonal rotation method

minimizes the Crawford-Ferguson family objective with \(\kappa=\frac{k-1}{p+k-2}\)

parsimonyorthogonal rotation method

minimizes the Crawford-Ferguson family objective with \(\kappa=1\)

quartiminoblique rotation method that

minimizes the oblimin objective with \(\gamma=0\)

biquartiminoblique rotation method that

minimizes the oblimin objective with \(\gamma=\frac{1}{2}\)

targetorthogonal or oblique rotation that rotates towards a target

matrix \(H\) by minimizing the objective

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

method_args:

Hndarray

target matrix

rotation_method{‘orthogonal’, ‘oblique’}

should be one of {orthogonal, oblique}

For orthogonal rotations the algorithm can be set to analytic in which case the following keyword arguments are available:

full_rankbool, optional

if set to true full rank is assumed

partial_target : orthogonal (default) or oblique rotation that partially rotates towards a target matrix \(H\) by minimizing the objective:

\[\phi(L) =\frac{1}{2}\|W\circ(L-H)\|^2.\]

method_args:

Hndarray

target matrix

Wndarray, optional

matrix with weights, entries can either be one or zero. The default is a matrix of ones, i.e., equal weight for all entries.

Examples

>>> import numpy as np
>>> A = np.random.randn(8,2)
>>> L, T = rotate_factors(A,'varimax')
>>> np.allclose(L,A.dot(T))
True
>>> L, T = rotate_factors(A,'orthomax',0.5)
>>> np.allclose(L,A.dot(T))
True
>>> L, T = rotate_factors(A,'quartimin',0.5)
>>> np.allclose(L,A.dot(np.linalg.inv(T.T)))
True