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psychometrikavol. 80, no. 3, 776790 September 2015doi: 10.1007/s11336-014-9416-y
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Web End = SPARSE VERSUS SIMPLE STRUCTURE LOADINGS
Nickolay T. Trendafilov
OPEN UNIVERSITY
Kohei Adachi
OSAKA UNIVERSITY
The component loadings are interpreted by considering their magnitudes, which indicates how strongly each of the original variables relates to the corresponding principal component. The usual ad hoc practice in the interpretation process is to ignore the variables with small absolute loadings or set to zero loadings smaller than some threshold value. This, in fact, makes the component loadings sparse in an articial and a subjective way. We propose a new alternative approach, which produces sparse loadings in an optimal way. The introduced approach is illustrated on two well-known data sets and compared to the existing rotation methods.
Key words: principal component analysis, factor analysis, orthogonal and oblique rotations, sparseness-inducing constraints, LASSO constraints, projected gradients..
1. Introduction
The goal of the simple structure approach is to transform some initial component loadings matrix into a new one having simple structure, and thus is easier to interpret. The adopted transformations are orthogonal or oblique rotations, which give the name rotation methods. Originally, the simple structure concept was introduced in factor analysis (FA) as three rules in the 30s of the last century, and was later extended and elaborated (Thurstone, 1947, p. 335). In formal terms, the simple structure approach nds a rotation, which optimizes certain objective function dening the Thurstones simple structure concept as a single formula. Later, the simple structure approach was adapted to principal component analysis (PCA) to make the rotated component loadings as interpretable as possible (Jolliffe, 2002, Chap. 11).
We stress on the fact that the original simple structure concept requires vanishing and, in fact, sparse loadings. The term sparse loadings was introduced by Zou, Hastie, and Tibshirani (2006) and means a matrix of component loadings containing a lot of zero entries. Unfortunately, true sparseness has never been achieved by the classical rotation methods.
We propose a new alternative approach, which produces sparse loadings in an optimal way. The introduced approach is illustrated on two well-known data sets and compared to the existing rotation methods.
2. Rotated Component Loadings
The rotation methods have a long...