naginterfaces.library.mv.rot_​promax

naginterfaces.library.mv.rot_promax(stand, x, ro, power, io_manager=None)

rot_promax calculates a ProMax rotation, given information following an orthogonal rotation.

For full information please refer to the NAG Library document for g03bd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g03/g03bdf.html

Parameters

standstr, length 1

Indicates how loadings are normalized.

$stand=\text{'S'}$

Rows of $Y$ are (Kaiser) normalized by the communalities of the loadings.

$stand=\text{'U'}$

Rows are not normalized.

xfloat, array-like, shape $(n,m)$

The loadings matrix following an orthogonal rotation, $X$.

rofloat, array-like, shape $(m,m)$

The orthogonal rotation matrix, $O$.

powerfloat

$p$, the value of exponent.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

fpfloat, ndarray, shape $(n,m)$

The factor pattern matrix, $P$.

rfloat, ndarray, shape $(m,m)$

The ProMax rotation matrix, $R$.

phifloat, ndarray, shape $(m,m)$

The matrix of inter-factor correlations, $\Phi$.

fsfloat, ndarray, shape $(n,m)$

The factor structure matrix, $S$.

Raises

NagValueError

(errno $1$)

On entry, $stand=⟨value⟩$.

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

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $1$)

On entry, $power=⟨value⟩$.

Constraint: $power>1.0$.

(errno $2$)

On entry, $n=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $n\ge m$.

(errno $20$)

The singular value decomposition has failed to converge.

(errno $100$)

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

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 $X$ and $Y$ denote $n\times m$ matrices each representing a set of $n$ points in an $m$-dimensional space. The $X$ matrix is a matrix of loadings as returned by rot_orthomax(), that is following an orthogonal rotation of a loadings matrix $Z$. The target matrix $Y$ is calculated as a power transformation of $X$ that preserves the sign of the loadings. Let $X_{ij}$ and $Y_{ij}$ denote the $(i,j)$th element of matrices $X$ and $Y$. Given a value greater than $1$ for the exponent $p$:

$$Y_{ij}={\delta }_{ij}{\Vert X_{ij}\Vert }^p\text{,}$$

for

$i=1,2,\dots ,n$;

$j=1,2,\dots ,m$;

${\delta }_{ij}=\{\begin{array}{c}-1\text{, if}{X}_{ij}<0\text{;}\\ 1\text{, otherwise.}\end{array}$

The above power transformation tends to increase the difference between high and low values of loadings and is intended to increase the interpretability of a solution.

In the second step a solution of:

$$XW=Y\text{,}X,Y\in R^{n\times m}\text{,}W\in R^{m\times m}\text{,}$$

is found for $W$ in the least squares sense by use of singular value decomposition of the orthogonal loadings $X$. The ProMax rotation matrix $R$ is then given by

$$R=OW\tilde{W}\text{,}O\text{,}\tilde{W}\in R^{m\times m}\text{,}$$

where $O$ is the rotation matrix from an orthogonal transformation, and $\tilde{W}$ is a matrix with the square root of diagonal elements of ${\left(W^{T}W\right)}^{-1}$ on its diagonal and zeros elsewhere.

The ProMax factor pattern matrix $P$ is given by

$$P=XW\tilde{W}\text{,}P\in R^{n\times m}\text{;}$$

the inter-factor correlations $\Phi$ are given by

$$\Phi ={\left(Q^{T}Q\right)}^{-1}\text{,}\Phi \in R^{m\times m}\text{;}$$

where $Q=W\tilde{W}$; and the factor structure $S$ is given by

$$S=P\Phi \text{,}S\in R^{n\times m}\text{.}$$

Optionally, the rows of target matrix $Y$ can be scaled by the communalities of loadings.