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:
  • A (numpy matrix (default None)) – non rotated factors
  • method (string) – should be one of the methods listed below
  • method_args (list) – additional arguments that should be provided with each method
  • algorithm_kwargs (dictionary) –
    algorithm : string (default gpa)
    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_tries : integer (default 501)
    maximum number of iterations
    tol : float
    stop criterion, algorithm stops if Frobenius norm of gradient is smaller then tol

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

    See the lower level functions for more details.
Returns:
Return type:

The tuple \((L,T)\)

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.
oblimin : orthogonal 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 rotations:

  • \(\gamma=0\) corresponds to quartimin,
  • \(\gamma=\frac{1}{2}\) corresponds to biquartimin.

method_args:

gamma : float
oblimin family parameter
rotation_method : string
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:

gamma : float (between 0 and 1)
orthomax family parameter

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:

kappa : float (between 0 and 1)
Crawford-Ferguson family parameter
rotation_method : string
should be one of {orthogonal, oblique}
quartimax : orthogonal rotation method
minimizes the orthomax objective with \(\gamma=0\)
biquartimax : orthogonal rotation method
minimizes the orthomax objective with \(\gamma=\frac{1}{2}\)
varimax : orthogonal rotation method
minimizes the orthomax objective with \(\gamma=1\)
equamax : orthogonal rotation method
minimizes the orthomax objective with \(\gamma=\frac{1}{p}\)
parsimax : orthogonal rotation method
minimizes the Crawford-Ferguson family objective with \(\kappa=\frac{k-1}{p+k-2}\)
parsimony : orthogonal rotation method
minimizes the Crawford-Ferguson family objective with \(\kappa=1\)
quartimin : oblique rotation method that minimizes
minimizes the oblimin objective with \(\gamma=0\)
quartimin : oblique rotation method that minimizes
minimizes the oblimin objective with \(\gamma=\frac{1}{2}\)

target : orthogonal or oblique rotation that rotates towards a target

matrix : math:H by minimizing the objective

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

method_args:

H : numpy matrix
target matrix
rotation_method : string
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_rank : boolean (default False)
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:

H : numpy matrix
target matrix
W : numpy matrix (default matrix with equal weight one for all entries)
matrix with weights, entries can either be one or zero

Examples

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