Page 281 - Contributed Paper Session (CPS) - Volume 7

CPS2099 Takatsugu Yoshioka et al.
2.2 Optimal scaling with alternating least squares
NLPCA reveals nonlinear relationships among variables with different
measurement levels and therefore presents a more flexible alternative to
ordinary CA.
Let X consist of categorical variables, each of which have

∗
categories ( = 1, . . . , ). Let be an optimal scaled matrix of data, i.e., -th
∗
optimal scaled variable = , where is × indicator matrix, is

× category quantifications of variable . NLPCA can find solutions by

minimizing two types of loss functions, a low-rank approximation and
homogeneity analysis with restrictions. The former loss function is
∗
∗
2
(, , ) = ‖ − ‖ , (2)
Where Z is × object scores. The minimization of loss functions has to
take place with respect to both parameters. The ALS algorithm is utilized to
solve such minimization problem. We describe the general procedure of an
ALS algorithm, PRINCIPALS (Young et al., 1978) that minimizes the loss
function (2). PRINCIPALS accepts nominal, ordinal and numerical variables,
and alternates between two estimation steps. The first step estimates the
model parameters Z and A for ordinary PCA, and the second obtains the
estimate of the data parameter for optimally scaled data. Given the initial
∗
data ∗(0) , PRINCIPALS iterates the following two steps:
Model estimation step:
By solving the Eigen-decomposition of ∗()T ∗() / or the singular value

decomposition of ∗() , obtain (+1) and compute (+1) =
̂

∗() (+1) .Update (+1)=(+1)(+1)T. (+1) = (+1) (+1)T
Optimal scaling step:
∗
Obtain ∗(+1) by separately estimating for each variable . Compute for

−1 (+1)

nominal variables as (+1) = ( ) ̂ . Re-compute (+1) for ordinal

variables using the monotone regression (Kruskal,1964). For nominal and
ordinal variables, update ∗(+1) = (+1) and standardize ∗(+1) . For

numerical variables, standardize observed vector and set ∗(+1) = .

2.3 RKM with NLPCA
We here propose a RKM with NLPCA based on RKM and NLPCA. Roughly
speaking, RKM with NLPCA is a method obtained by replacing dimension
reduction procedure (PCA step) in ordinary RKM by NLPCA which provides
both quantification and dimension reduction. The loss function is
2
∗
/ (, , )=‖ − ‖ (3)
The algorithm is as follows:
[Step1] Initialization: Determine the desired number of cluster and
components , considering the relation ≥+1. If you want the reasonable
number of , you may apply NLPCA to the data once. Assign random values
to U.

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