This function is a helper function of sma().
It performs polar docomposition, orthogonal rotation, and soft-thresholding shrinkage in order.
The three steps together enable sparse estimates of the SMA and SCA.
prs(x, z.hat, gamma, rotate, shrink, normalize, order, flip, epsilon)a matrix of the sparse estimate, of the same dimension as crossprod(x, z.hat).
the matrix product crossprod(x, z.hat) is the actual Polar-Rotate-Shrink object. x and z.hat are input separatedly because the former is additionally used to compute the proportion of variance explained, in the case when order = TRUE.
numeric, the sparsity parameter.
character(1), rotation method. Two options are currently available: "varimax" (default) or "absmin" (see details).
character(1), shrinkage method, either "soft"- (default) or "hard"-thresholding (see details).
logical, whether to rows normalization should be done before and undone afterward the rotation (see details).
logical, whether to re-order the columns of the estimates (see Details below).
logical, whether to flip the signs of the columns of estimates such that all columns are positive-skewed (see details).
numeric, tolerance of convergence precision (default to 0.00001).
rotate: The rotate option specifies the rotation technique to
use. Currently, there are two build-in options—“varimax” and “absmin”.
The “varimax” rotation maximizes the element-wise L4 norm of the rotated
matrix. It is faster and computationally more stable. The “absmin”
rotation minimizes the absolute sum of the rotated matrix. It is sharper
(as it directly minimizes the L1 norm) but slower and computationally
less stable.
shrink: The shrink option specifies the shrinkage operator to
use. Currently, there are two build-in options—“soft”- and
“hard”-thresholding. The “soft”-thresholding universally reduce all
elements and sets the small elements to zeros. The “hard”-thresholding
only sets the small elements to zeros.
normalize: The argument normalize gives an indication of if and
how any normalization should be done before rotation, and then undone
after rotation. If normalize is FALSE (the default) no normalization
is done. If normalize is TRUE then Kaiser normalization is done. (So
squared row entries of normalized x sum to 1.0. This is sometimes
called Horst normalization.) For rotate="absmin", if normalize is a
vector of length equal to the number of indicators (i.e., the number of
rows of x), then the columns are divided by normalize before
rotation and multiplied by normalize after rotation. Also, If
normalize is a function then it should take x as an argument and
return a vector which is used like the vector above.
order: In PCA (and SVD), the principal components (and the
singular vectors) are ordered. For this, we order the sparse components
(i.e., the columns of z or y) by their explained variance in the
data, which is defined as sum((x %*% y)^2), where y is a column of the
sparse component. Note: not to be confused with the cumulative
proportion of variance explained by y (and z), particularly when y
(and z) is may not be strictly orthogonal.
flip: The argument flip gives an indication of if and the
columns of estimated sparse component should be flipped. Note that the
estimated (sparse) loadings, i.e., the weights on original variables,
are column-wise invariant to a sign flipping. This is because flipping
of a principal direction does not influence the amount of the explained
variance by the component. If flip=TRUE, then the columns of loadings
will be flip accordingly, such that each column is positive-skewed. This
means that for each column, the sum of cubic elements (i.e., sum(x^3))
are non-negative.
Chen, F. and Rohe, K. (2020) "A New Basis for Sparse Principal Component Analysis."
sma, sca, polar, rotation, shrinkage