naginterfaces.library.mv.rot_​orthomax

naginterfaces.library.mv.rot_orthomax(stand, g, fl, acc=1e-05, maxit=30)

rot_orthomax computes orthogonal rotations for a matrix of loadings using a generalized orthomax criterion.

For full information please refer to the NAG Library document for g03ba

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g03/g03baf.html

Parameters

standstr, length 1

Indicates if the matrix of loadings is to be row standardized before rotation.

$stand=\text{'S'}$

The loadings are row standardized.

$stand=\text{'U'}$

The loadings are left unstandardized.

gfloat

$\gamma$, the criterion constant with $\gamma =1.0$ giving varimax rotations and $\gamma =0.0$ giving quartimax rotations.

flfloat, array-like, shape $(\text{nvar},k)$

$\Lambda$, the matrix of loadings. $fl[\text{i}-1,\text{j}-1]$ must contain the loading for the $\text{i}$th variable on the $\text{j}$th factor, for $\text{j}=1,2,\dots ,k$, for $\text{i}=1,2,\dots ,p$.

accfloat, optional

Indicates the accuracy required. The iterative procedure of Cooley and Lohnes (1971) will be stopped and the final refinement computed when the change in $V$ is less than $acc\times max(1.0,V)$. If $acc$ is greater than or equal to $0.0$ but less than machine precision or if $acc$ is greater than $1.0$, machine precision will be used instead.

maxitint, optional

The maximum number of iterations.

Returns

flfloat, ndarray, shape $(\text{nvar},k)$

If $stand=\text{'S'}$, the elements of $fl$ are standardized so that the sum of squared elements for each row is $1.0$ and then after the computation of the rotations are rescaled; this may lead to slight differences between the input and output values of $fl$.

If $stand=\text{'U'}$, $fl$ will be unchanged on exit.

flrfloat, ndarray, shape $(\text{nvar},k)$

The rotated matrix of loadings, ${\Lambda }^{\ast }$. $flr[\text{i}-1,\text{j}-1]$ will contain the rotated loading for the $\text{i}$th variable on the $\text{j}$th factor, for $\text{j}=1,2,\dots ,k$, for $\text{i}=1,2,\dots ,p$.

rfloat, ndarray, shape $(k,k)$

The matrix of rotations, $R$.

iteraint

The number of iterations performed.

Raises

NagValueError

(errno $1$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit>0$.

(errno $1$)

On entry, $\text{nvar}=⟨value⟩$ and $k=⟨value⟩$.

Constraint: $\text{nvar}\ge k$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 2$.

(errno $1$)

On entry, $g=⟨value⟩$.

Constraint: $g\ge 0.0$.

(errno $1$)

On entry, $stand=⟨value⟩$.

Constraint: $stand=\text{'S'}$ or $\text{'U'}$.

(errno $1$)

On entry, $acc=⟨value⟩$.

Constraint: $acc\ge 0.0$.

(errno $2$)

The singular value decomposition has failed to converge.

(errno $3$)

The algorithm to find R has failed to reach the required accuracy in $maxit$ iterations.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

Let $\Lambda$ be the $p\times k$ matrix of loadings from a variable-directed multivariate method, e.g., canonical variate analysis or factor analysis. This matrix represents the relationship between the original $p$ variables and the $k$ orthogonal linear combinations of these variables, the canonical variates or factors. The latter are only unique up to a rotation in the $k$-dimensional space they define. A rotation can then be found that simplifies the structure of the matrix of loadings, and hence the relationship between the original and the derived variables. That is, the elements, ${\lambda }_{ij}^{\ast }$, of the rotated matrix, ${\Lambda }^{\ast }$, are either relatively large or small. The rotations may be found by minimizing the criterion:

$$V=\sum _{j=1}^k\sum _{i=1}^p{\left({\lambda }_{ij}^{\ast }\right)}^4-\frac{\gamma }{p}\sum _{j=1}^k{\left[\sum _{i=1}^p{\left({\lambda }_{ij}^{\ast }\right)}^2\right]}^2$$

where the constant $\gamma$ gives a family of rotations with $\gamma =1$ giving varimax rotations and $\gamma =0$ giving quartimax rotations.

It is generally advised that factor loadings should be standardized, so that the sum of squared elements for each row is one, before computing the rotations.

The matrix of rotations, $R$, such that ${\Lambda }^{\ast }=\Lambda R$, is computed using first an algorithm based on that described by Cooley and Lohnes (1971), which involves the pairwise rotation of the factors. Then a final refinement is made using a method similar to that described by Lawley and Maxwell (1971), but instead of the eigenvalue decomposition, the algorithm has been adapted to incorporate a singular value decomposition.

References

Cooley, W C and Lohnes, P R, 1971, Multivariate Data Analysis, Wiley

Lawley, D N and Maxwell, A E, 1971, Factor Analysis as a Statistical Method, (2nd Edition), Butterworths