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CPS1950 Paolo G. et al.
As for RKM, parameters can be addressed by the least square estimation
method. In particular, the optimal parameter matrices are found by minimizing
2
fFKM = || E || = || XAA  UFA || = || XA  UF || . (4)
2
2
For this purpose, an ALS algorithm can be used. The FKM solution is found up
to rotational indeterminacy for the weights and the centroids and cluster label
switching. If A = IJ, FKM coincides with KM.
c. Comparison between FKM and RKM:
Vichi & Kiers (2001) and Timmerman et al. (2013) investigate the FKM and
RKM procedures from a theoretical and a practical point of view. Their findings
are summarized in this subsection. First of all, Vichi & Kiers (2001) show that
RKM may fail when a large amount of variance pertains to directions
orthogonal to the one relevant for clustering purposes. It implicitly suggests
that the two methods model the data in different ways. Timmerman et al.
(2013) extensively analyze this point by defining two types of residuals,
namely, subspace residuals and complement residuals. The former ones
denote the residuals lying on the subspace spanned by the columns of A. The
latter ones refer to the residuals lying on the complement of this subspace,
i.e., those lying on the subspace spanned by the columns of A , being A a
┴
┴
┴
column-wise orthonormal matrix of order (J × J  Q) such that AA = 0Q × J 
Q, where 0Q × J  Q is the matrix of zeroes of order (Q × J  Q). Real life data
usually contain both kinds of residuals. The performances in recovering the
clusters of FKM and RKM are related to the relative sizes of such two kinds of
residuals. Specifically, FKM performs better than RKM when the complement
residuals are smaller than the subspace ones. Conversely, RKM outperforms
FKM when the subspace residuals are small if compared to the complement
ones.
d. Fuzzy Reduced and Factorial K-Means (FRFKM):
Taking into account that the objectives of FKM and RKM are different and
every method aims at minimizing a specific kind of residuals, our idea is to
exploit the potentialities of both methods by considering them
simultaneously. In doing so, we adopt the fuzzy approach to clustering by
relaxing the constraints that the elements of U are either 0 or 1. In order to
handle the concept of partial truth, where the truth value may range between
completely false and completely true, it implies that every observation unit can
be assigned to a certain cluster with the so-called fuzzy membership degree
ranging from 0 (non-membership, completely false) to 1 (full membership,
completely true).
The new procedure, called Fuzzy Reduced and Factorial K-Means (FRFKM),
is developed by considering a linear convex combination of the loss functions
in fRKM in (2) and fFKM in (4). Thus, we have
fFRFKM = (1)fRKM + fFKM, (5)

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