Constrained principal component analysis of standardized data for biplots with unit-length variable vectors

Adachi, Keiichiro

doi:10.1007/s11634-010-0081-4

Cited by 6 publications

(3 citation statements)

References 12 publications

(8 reference statements)

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“…is, the average of the elements in 2 −1 (M+M) (Adachi, 2011). Index (29) takes a value of zero to one and attains one forM = M. Table 1 shows the quartiles of the standardized similarities for loadings, unique variances, and factor scores, i.e., SS(Λ, Λ), SS(Ψ 2 1 p , Ψ 2 1 p ), SS(F, F), and SS(Û, U), respectively, for the solutions of each procedure.…”

Section: Resultsmentioning

confidence: 99%

SOME CONTRIBUTIONS TO DATA-FITTING FACTOR ANALYSIS WITH EMPIRICAL COMPARISONS TO COVARIANCE-FITTING FACTOR ANALYSIS

Adachi

2012

Journal of the Japanese Society of Computational Statistics

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A data-fitting factor analysis (FA) procedure was recently presented, which is very different from the prevailing covariance-fitting FA. In the former procedure, common and unique factor scores are modeled as fixed unknown parameters, and an unweighted least squares (ULS) function, which is not scale invariant, is minimized for fitting the model to a data matrix. The main purpose of this paper is to settle four remaining problems with data-fitting FA. First, we present a weighted least squares (WLS) procedure which can be scale invariant, and include the above ULS procedure as a special case according to the choice of weights. Second, we prove that the WLS loss function can be minimized, even if raw data are unknown and only their sample covariance matrix is available, despite being a data-fitting approach. Third, we propose an estimator of factor scores that cannot be uniquely determined. Fourth, we empirically compare this data-fitting FA procedure with covariance-fitting FA with respect to recovery of parameter matrices.

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Section: Resultsmentioning

confidence: 99%

SOME CONTRIBUTIONS TO DATA-FITTING FACTOR ANALYSIS WITH EMPIRICAL COMPARISONS TO COVARIANCE-FITTING FACTOR ANALYSIS

Adachi

2012

Journal of the Japanese Society of Computational Statistics

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“…To avoid the local minimum and obtain a solution as close as possible to the global minimizer, we use the procedure proposed by Adachi (2011) in which the FCBA algorithm is run multiple times starting from different initial values until the two equivalently best solutions are found. By using S m = {U m , V m , P m , Q m } for the solution obtained by the m-th run and f FC B (S m ) for the attained value of f FC B (U m , V m , P m , Q m ), the multiple runs procedure for FCBA is stated as follows:…”

Section: Multiple Runs Proceduresmentioning

confidence: 99%

“…. ., a (J ) } , the data matrix X was constructed with X = FA + θ(ρ)E, where E was filled with standard normal errors and θ(ρ) given as follows (Adachi 2011):…”

Section: Simulation Studymentioning

confidence: 99%

A new biplot procedure with joint classification of objects and variables by fuzzy c-means clustering

Yamashita

Mayekawa

2014

Adv Data Anal Classif

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Biplot is a technique for obtaining a low-dimensional configuration of the data matrix in which both the objects and the variables of the data matrix are jointly represented as points and vectors, respectively. However, biplots with a large number of objects and variables remain difficult to interpret. Therefore, in this research, we propose a new biplot procedure that allows us to interpret a large data matrix. In particular, the objects and variables are classified into a small number of clusters by using fuzzy c-means clustering and the resulting clusters are simultaneously biplotted in lower-dimensional space. This procedure allows us to understand the configurations easily and to grasp the fuzzy memberships of the objects and variables to the clusters. A simulation study and real data example are also provided to demonstrate the effectiveness of the proposed procedure.

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Layered Multivariate Regression with Its Applications

Yamashita

Adachi

2020

Studies in Classification, Data Analysis, and Knowledge Organization

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Constrained principal component analysis of standardized data for biplots with unit-length variable vectors

Abstract: Biplots, Unit-length vector analysis , Principal component analysis, Inner product, Projection, 62H25, 91C15, 15A99,

Cited by 6 publications

References 12 publications

<b>SOME CONTRIBUTIONS TO DATA-FITTING FACTOR </b><b>ANALYSIS WITH EMPIRICAL COMPARISONS TO </b><b>COVARIANCE-FITTING FACTOR ANALYSIS </b>

<b>SOME CONTRIBUTIONS TO DATA-FITTING FACTOR </b><b>ANALYSIS WITH EMPIRICAL COMPARISONS TO </b><b>COVARIANCE-FITTING FACTOR ANALYSIS </b>

A new biplot procedure with joint classification of objects and variables by fuzzy c-means clustering

Layered Multivariate Regression with Its Applications

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