A set of non-negatively correlated variables,
referred to as positive manifold data, display a
peculiar pattern of loadings in principal components
analysis (PCA). If a small set of principal
components is rotated to a simple structure, the
variables correlate positively with all components,
thus displaying positive manifold. However, this
phenomenon is critically dependent on the freedom
of rotation, as is evident from the unrotated
loadings. That is, although the first principal
component is without contrast (which means that
all variables correlate either positively or negatively
with the first component), subsequent components
have mixtures of positive and negative loadings-which means that positive manifold is absent.
PARAFAC is a generalization of PCA that has unique
components, which means that rotations are not
allowed. This paper examines how PARAFAC
behaves when applied to positive manifold data. It
is shown that PARAFAC does not always produce
positive manifold solutions. For cases in which
PARAFAC does not produce a positive manifold
solution, a constrained PARAFAC method is offered
that restores positive manifold by introducing non-negativity
constraints. Thus, noncontrast PARAFAC
components can be found that explain only a
negligible amount of variance less than the
PARAFAC components. These noncontrast components
cannot be degenerate and cannot be partially
unique in the traditional sense. Index terms:
degenerate components; noncontrast components; non-negativity
constraints; PARAFAC; positive manifold.
Krijnen, Wim P & ten Berge, Jos M. (1992). A constrained PARAFAC method for positive manifold data. Applied Psychological Measurement, 16, 295-305. doi:10.1177/014662169201600310
Krijnen, Wim P.; Ten Berge, Jos M. F..
A constrained PARAFAC method for positive manifold data.
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