TITLE:
Two-Stage Procrustes Rotation with Sparse Target Matrix and Least Squares Criterion with Regularization and Generalized Weighting
AUTHORS:
Naoto Yamashita
KEYWORDS:
Factor Rotation, Procrustes Rotation, Simplicity, Alternating Least Squares
JOURNAL NAME:
Open Journal of Statistics,
Vol.13 No.2,
April
27,
2023
ABSTRACT: In factor analysis, a factor loading matrix is often rotated to a simple
target matrix for its simplicity. For the purpose, Procrustes rotation
minimizes the discrepancy between the target and rotated loadings using two
types of approximation: 1) approximate the zeros in the target by the non-zeros
in the loadings, and 2) approximate the non-zeros in the target by the
non-zeros in the loadings. The central issue of Procrustes rotation considered
in the article is that it equally treats the two types of approximation, while
the former is more important for simplifying
the loading matrix. Furthermore, a well-known issue of Simplimax is the
computational inefficiency in estimating the sparse target matrix, which yields
a considerable number of local minima. The research proposes a new rotation
procedure that consists of the following two stages. The first stage estimates
sparse target matrix with lesser computational cost by regularization
technique. In the second stage, a loading matrix is rotated to the target,
emphasizing on the approximation of non-zeros to zeros in the target by least
squares criterion with generalized weighing that is newly proposed by the
study. The simulation study and real data examples revealed that the proposed
method surely simplifies loading matrices.