Exploratory Procedure for Component-Based Structural Equation Modeling for Simple Structure by Simultaneous Rotation

Naoto Yamashita

doi:10.1007/s11336-023-09942-5

Exploratory Procedure for Component-Based Structural Equation Modeling for Simple Structure by Simultaneous Rotation

Published online by Cambridge University Press: 27 December 2024

Naoto Yamashita

Show author details

Naoto Yamashita*: Affiliation:
Kansai University
*: Correspondence should be made to Naoto Yamashita, Faculty of Sociology, Kansai University, Osaka, Japan. Email: [email protected] https://sites.google.com/site/nyamashitahp

Article contents

Abstract
Introduction
Proposed Method
Numerical Simulation
Real Data Example
Conclusion
Funding
Declarations
Footnotes
References

Rights & Permissions

Abstract

Generalized structured component analysis (GSCA) is a structural equation modeling (SEM) procedure that constructs components by weighted sums of observed variables and confirmatorily examines their regressional relationship. The research proposes an exploratory version of GSCA, called exploratory GSCA (EGSCA). EGSCA is analogous to exploratory SEM (ESEM) developed as an exploratory factor-based SEM procedure, which seeks the relationships between the observed variables and the components by orthogonal rotation of the parameter matrices. The indeterminacy of orthogonal rotation in GSCA is first shown as a theoretical support of the proposed method. The whole EGSCA procedure is then presented, together with a new rotational algorithm specialized to EGSCA, which aims at simultaneous simplification of all parameter matrices. Two numerical simulation studies revealed that EGSCA with the following rotation successfully recovered the true values of the parameter matrices and was superior to the existing GSCA procedure. EGSCA was applied to two real datasets, and the model suggested by the EGSCA’s result was shown to be better than the model proposed by previous research, which demonstrates the effectiveness of EGSCA in model exploration.

Keywords

structural equation modeling generalized structured component analysis exploratory data analysis orthogonal rotation

Type: Theory and Methods
Information: Psychometrika , Volume 89 , Issue 2 , June 2024 , pp. 411 - 438

DOI: https://doi.org/10.1007/s11336-023-09942-5 [Opens in a new window]
Copyright: Copyright © 2023 The Author(s) under exclusive licence to The Psychometric Society

Introduction

Structural equation modeling (SEM) (Bentler, Reference Bentler1980, Reference Bentler1986; Mulaik, Reference Mulaik1986) is widely used across various fields including Psychology and Sociology, as a method for examining causal structure between observed and latent variables. There are several procedures that enable the concept of SEM and they are classified into the following two approaches.

The first is factor-based SEM, and the procedure implemented in the LISREL software package by Jöreskog and Sörbom (Reference Jöreskog and Sörbom1996) is most popular. Factor-based SEM describes the relationship between the observed and latent variables as a form of measurement equation, whereas the causal relationships between factors are described by structural equations. The model parameters are estimated by fitting the covariance structure derived from the series of equations to the observed covariance matrix.

The second approach is component-based SEM, where the latent variables are defined as the weighted sum of the observed variables (Tenenhaus, Reference Tenenhaus2008), whereas they are defined as a random variable in factor-based SEM. The weighted sum should be called a component rather than a factor. Hwang and Takane (Reference Hwang and Takane2004) proposed a procedure for component-based SEM called generalized structured component analysis (GSCA). GSCA was originally proposed as an elaboration of partial least squares path modeling (PLS-PM; Tenenhaus et al. Reference Tenenhaus, Vinzi, Chatelin and Lauro2005; Wold et al. Reference Wold, Sjmöstrmöm and Eriksson2001), and its special case called PLS regression (Esposito Vinzi and Russolillo, Reference Esposito Vinzi and Russolillo2013) is frequently used in Chemometrics. GSCA’s objective is to minimize the least squares criterion given an $N \times J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \times J$$\end{document} data matrix $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} of N samples observed by J observed variables

(1)

\begin{matrix} f_{GSCA} (W, C, B) = | | Z [I_{J}, W] - ZW [C, B] {| |}^{2} \end{matrix}

where $W$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}$$\end{document} is the $J \times D$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J\times D$$\end{document} matrix of component weights for D components associated with J variables and $I_{J}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{I}_J$$\end{document} is the J-dimensional identity matrix. $C (D \times J)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}(D\times J)$$\end{document} and $B (D \times D)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}(D\times D)$$\end{document} are matrices of component loadings and path coefficients, respectively. In GSCA, the D components are formed by J variables using the weight matrix $W$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}$$\end{document} as $Γ = ZW$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Gamma }} = \textbf{ZW}$$\end{document} . The criterion in (1) is rewritten as

(2)

\begin{matrix} f_{GSCA} (W, C, B) = | | Z - {ZWC | |}^{2} + | | ZW - {ZWB | |}^{2} \end{matrix}

indicating the following twofold aim of GSCA: The original J variables in $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} are regressed on their component scores $ZW$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{ZW}$$\end{document} with the loading matrix $C$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}$$\end{document} , and the component scores are regressed on themselves with the matrix of the path coefficients between D components noted as $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}$$\end{document} . The least squares criterion in (1) is minimized under the constraint

(3)

\begin{matrix} diag (Γ^{'} Γ) = I_{D} \end{matrix}

meaning that the component scores have a unit variance. Some extensions of GSCA have been proposed: regularization of parameter matrices (Hwang, Reference Hwang2009), combination with fuzzy clustering (Hwang et al., Reference Hwang, Desarbo and Takane2007), parameterization of measurement error (Hwang et al., Reference Hwang, Kim, Lee and Park2017a, Reference Hwang, Takane and Jungb), and the integration with factor-based SEM (Hwang et al., Reference Hwang, Cho, Jung, Falk, Flake, Jin and Lee2021).

The critical difference between GSCA and PLS-PM is that according to Hwang et al. (Reference Hwang and Takane2014), the former is formulated as a minimization problem of the least squares criterion defined in (1), while the latter does not optimize a single criterion. This leads that the global goodness-of-fit cannot be measured in PLS-PM, and the convergence of repeated computation in PLS-PM does not have a clear meaning.

In GSCA, the hypothetical relationships between observed variables and components are described by restricting some elements of the parameter matrices to zero. For example, consider the path diagram shown in Fig. 1 where the nine observed variables noted as $z_{1}, \dots, z_{9}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}_1,\ldots ,\textbf{z}_9$$\end{document} form the four components noted as $γ_{1}, \dots, γ_{4}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\gamma }}_1,\ldots ,{\varvec{\gamma }}_4$$\end{document} . The paths from the variables to the components are expressed by $W$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}$$\end{document} , whereas the paths in the opposite direction are noted as $C$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}$$\end{document} . Based on the path diagram, some of the elements in $W$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}$$\end{document} and $C$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}$$\end{document} are fixed at 0, and the others are treated as free parameters

(4)

\begin{matrix} W = (\begin{matrix} w_{1} & 0 & 0 & 0 \\ w_{2} & 0 & 0 & 0 \\ w_{3} & 0 & 0 & 0 \\ 0 & w_{4} & 0 & 0 \\ 0 & w_{5} & 0 & 0 \\ 0 & 0 & w_{6} & 0 \\ 0 & 0 & w_{7} & 0 \\ 0 & 0 & 0 & w_{8} \\ 0 & 0 & 0 & w_{9} \end{matrix}), C = (\begin{matrix} c_{1} & c_{2} & c_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{4} & c_{5} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{6} & c_{7} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{8} & c_{9} \end{matrix}) \end{matrix}

where $w_{1}, \dots, w_{9}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1,\ldots ,w_9$$\end{document} and $c_{1}, \dots, c_{9}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1,\ldots ,c_9$$\end{document} denote the free parameters to be estimated. Similarly, the regressional relationship between components is shown in $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}$$\end{document} :

(5)

\begin{matrix} B = (\begin{matrix} 0 & 0 & b_{13} & 0 \\ 0 & 0 & b_{23} & b_{24} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) . \end{matrix}

Figure 1 An example of path diagram for GSCA.

The above-mentioned two approaches for SEM including GSCA are considered to be confirmatory data analysis (CDA). Here, we refer to CDA as the family of analysis aimed at testing a hypotheses about the given data, as opposed to exploratory data analysis (EDA). Specifically, factor-/component-based techniques necessitate that users create hypotheses about the relationship between factors (components) and variables as well as the causal connections between the factors. For instance, GSCA represents a hypothesis by the pattern of unknown parameters in the parameter matrices $W$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}$$\end{document} , $C$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}$$\end{document} , and $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}$$\end{document} . There are many cases, however, where the hypothesis is unclear, such as the early stage of a study, and the existing SEM procedures are therefore inappropriate. Taking confirmatory factor analysis (CFA) which is the most prevalent CDA procedure, Marsh et al. (Reference Marsh, Morin, Parker and Kaur2014) pointed out that the oversimplified measurement model, which is often used in CFA, has the potential risk of misspecification of the model, and limiting some of the parameters to 0 may degrade the goodness-of-fit to the data. In practical use of CFA, when fitting a model, the fitted model is improved by referring to a goodness-of-fit index and a modification index, and the improved model is fitted again to the data, until the value of goodness-of-fit index is reasonably high. In other words, CFA requires the repeated process of model fitting and modification, which indicates that CDA often ends up as an exploratory process. Their argument is seen to be relevant to all CDA procedures including SEM and GSCA, which are the subject of the research.

Asparouhov and Muthén (Reference Asparouhov and Muthén2009) proposed the exploratory procedure for factor-based SEM, called exploratory SEM (ESEM). ESEM first fits the largest model which has as many free parameters as possible to the extent that the model is identified. Next, by exploiting rotational indeterminacy in factor-based SEM, ESEM rotates some of the parameter matrices and finds simple structure. Varimax (Kaiser, Reference Kaiser1958) and target rotation (Browne, Reference Browne1972a, Reference Browneb, Reference Browne2001) are often used for this purpose (Marsh et al., Reference Marsh, Morin, Parker and Kaur2014). Examples of recent applications are found in personality research (McLarnon, Reference McLarnon2022), school psychology (Pianta et al., Reference Pianta, Lipscomb and Ruzek2022), linguistics (Alamer, Reference Alamer2022), and consumer research (Poier et al., Reference Poier, Nikodemska-Wołowik and Suchanek2022). A further elaboration of ESEM was proposed by Marsh et al. (Reference Marsh, Guo, Dicke, Parker and Craven2020).

Despite the successful applications of ESEM, they are based on factor-based SEM, and an exploratory procedure for component-based SEM has not yet been proposed. To meet the potential needs of exploratory procedure in component-based SEM, this article first shows that the solutions in GSCA can be orthogonally rotated without loss of fit. Based on rotational indeterminacy, the study proposes an exploratory GSCA (EGSCA), which seeks simple relationships between observed variables and components. Various orthogonal rotational procedures such as Varimax can be employed for EGSCA. The existing rotational procedures simplify a single parameter matrix; however, EGSCA simultaneously rotates multiple parameter matrices. A new procedure for orthogonal rotation is also developed, where all parameter matrices in EGSCA are simultaneously simplified, and it is called a simultaneous target rotation.

As mentioned earlier, GSCA is preferred over PLS-PM because it optimizes a single optimization criterion, which makes GSCA more plausible than PLS-PM as a component-based SEM procedure. Additionally, this property ensures the indeterminacy of GSCA’s parameter matrices with respect to orthogonal rotation, providing theoretical support for EGSCA. On the other hand, the transformation is not permitted in PLS-PM, because it lacks an optimization criterion. As a result, the research employs GSCA instead of PLS-PM to develop an exploratory procedure for component-based SEM.

Hwang (Reference Hwang2009) proposed an extension of GSCA which is similar to EGSCA called regularized GSCA (RGSCA). It is designed to solve multi-collinearity problem in GSCA, where parameter estimation is unstable because the observed and latent variables are highly correlated. RGSCA is formulated as the minimization of penalized least squares criterion where the parameter matrices are regularized by $L_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} penalty

(6)

\begin{matrix} f_{GSCA} (W, C, B) + λ_{1} P (W) + λ_{2} P (B) + λ_{3} P (C) \end{matrix}

with $P (W)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\textbf{W})$$\end{document} being the $L_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} penalty defined as $P (W) = \sum_{j = 1}^{J} \sum_{d = 1}^{D} {| w_{jd} |}^{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(\textbf{W}) = \sum _{j=1}^{J}\sum _{d=1}^{D} |w_{jd}|^2$$\end{document} , and $λ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1$$\end{document} , $λ_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2$$\end{document} , and $λ_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _3$$\end{document} are positive integers that control the strength of the penalty. The regularization works to shrink the nonzero elements in all parameter matrices toward zero. Using this property, RGSCA can exploratorily extract the relationship between observed and latent variables. The exploration starts with the initial model expressed as a path diagram, and the model is further re-expressed as the positions of free parameters in the parameter matrices. The exploration using RGSCA is achieved by eliminating some of the paths of the prespecified path diagram, by shrinking the corresponding path coefficient toward zero. The scope of the exploration is therefore limited to the sub-models that are contained in the initial model, and this would lead to the model misspecification when the true model is not in the scope, as numerically demonstrated in the third section. The proposed EGSCA avoids the risk of model misspecification because it uses the largest model that contains the maximum number of parameters and exploits the simple relationships between the variables by rotation. The exploration scope is therefore wider than RGSCA, and a more plausible model can be found by EGSCA. In the numerical simulation, we empirically show that EGSCA is better than RGSCA in model exploration.

RGSCA often produces a solution that contains some elements close to zero, and EGSCA’s solution also contains such elements which are attained by orthogonal rotation. The critical difference between RGSCA and EGSCA is that the latter seeks a simple structure by rotating the parameter matrices, while RGSCA does not consider the simplicity of parameter matrices. To obtain insight into the given data for subsequent CDA, it is reasonable for the parameter matrices to have a simple structure, and therefore, EGSCA is suitable for an exploratory data analysis, as demonstrated in the fourth section.

In addition, the RGSCA solution varies when different values of tuning parameters $λ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1$$\end{document} , $λ_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2$$\end{document} , and $λ_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _3$$\end{document} are given, and therefore, they have to be chosen carefully by cross-validation, for example. The proposed procedure does not require such difficult tuning, however, which would be welcomed by potential users of EGSCA.

The remaining parts of this article are organized as follows. The second section shows the indeterminacy of orthogonal rotation in GSCA by reformulating GSCA’s mathematical formulation adding some additional constraints. The following subsection proposes EGSCA, as an exploratory variant of GSCA. Further, a new rotational procedure is proposed to simplify all the parameter matrices simultaneously, followed by the subsection which discuss the choice of hyperparameters in the rotation algorithm. The performance of the proposed procedure is evaluated by two numerical simulation studies in the third section. Two examples of EGSCA applied to real datasets are presented in the fourth section. The final section concludes the article.

2. Proposed Method

In the following four subsections, GSCA is shown to have the property that its parameter matrices are orthogonally rotated without loss of fit under some assumptions, followed by the proposal of an exploratory variant of GSCA named EGSCA. The fourth subsection discusses how the choice of the hyperparameters in the rotation algorithm affects the resulting simplicity and provides practical guidance for the choice.

2.1. Rotational Indeterminacy in GSCA

To show the rotational indeterminacy in GSCA, we begin by rewriting the least squares criterion in (1) using some block matrices. Here, consider that the J observed variables in $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} are divided into the two mutually exclusive groups, the group having $J_{1} (< J)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_1 (< J)$$\end{document} variables and the remaining $J_{2} = J - J_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_2 = J - J_1$$\end{document} variables for the other group. The partition of the observed variables is equivalent to the exogenous/endogenous groups often assumed in LISREL formulation (Jöreskog and Sörbom, Reference Jöreskog and Sörbom1996; Wang and Wang, Reference Wang and Wang2019). Let $Z_{1} (N \times J_{1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_1(N\times J_1)$$\end{document} and $Z_{2} (N \times J_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_2(N\times J_2)$$\end{document} be the matrices of $J_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_1$$\end{document} and $J_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_2$$\end{document} variables, respectively, and $Z = [Z_{1}, Z_{2}]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z} = [\textbf{Z}_1, \textbf{Z}_2]$$\end{document} . Further, assume that the two groups of observed variables form $D_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_1$$\end{document} and $D_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2$$\end{document} components with the weight matrices $W_{1} (J_{1} \times D_{1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1(J_1\times D_1)$$\end{document} and $W_{2} (J_{2} \times D_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2(J_2\times D_2)$$\end{document} , respectively. The above assumptions indicate that $W$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}$$\end{document} is expressed as a block-diagonal matrix

(7)

\begin{matrix} W = (\begin{matrix} W_{1} \\ W_{2} \end{matrix}) \end{matrix}

and $C$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}$$\end{document} is also expressed as

(8)

\begin{matrix} C = (\begin{matrix} C_{1} \\ C_{2} \end{matrix}) \end{matrix}

where $C_{1} (D_{1} \times J_{1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1(D_1\times J_1)$$\end{document} and $C_{2} (D_{2} \times J_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2(D_2\times J_2)$$\end{document} are loading matrices of the first and second groups of observed variables, respectively. $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}$$\end{document} is partitioned as

(9)

\begin{matrix} B = (\begin{matrix} {\tilde{B}}_{1} & B_{1} \\ B_{2} & {\tilde{B}}_{2} \end{matrix}) \end{matrix}

where ${\tilde{B}}_{1} (D_{1} \times D_{1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_1(D_1\times D_1)$$\end{document} and ${\tilde{B}}_{2} (D_{2} \times D_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_2(D_2\times D_2)$$\end{document} are the path coefficient matrices within the components formed by the variables $Z_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_1$$\end{document} and $Z_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_2$$\end{document} , respectively. $B_{1} (D_{1} \times D_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1(D_1\times D_2)$$\end{document} and $B_{2} (D_{2} \times D_{1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2(D_2\times D_1)$$\end{document} are the matrices of path coefficients between the two sets of components. It should be noted that the diagonal elements of ${\tilde{B}}_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_1$$\end{document} and ${\tilde{B}}_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_2$$\end{document} must be zero to suppress self-regression of each of the components.

Using the above notation, the least squares criterion of GSCA in (1) is rewritten as

(10)

\begin{matrix} f_{GSCA} (W, C, B) = & | | Z_{1} - Z_{1} W_{1} C_{1} {| |}^{2} + | | Z_{2} - Z_{2} W_{2} C_{2} {| |}^{2} \\ + | | Z_{1} W_{1} - (Z_{1} W_{1} {\tilde{B}}_{1} + Z_{2} W_{2} B_{2}) {| |}^{2} \\ + | | Z_{2} W_{2} - (Z_{1} W_{1} B_{1} + Z_{2} W_{2} {\tilde{B}}_{2}) {| |}^{2} \end{matrix}

indicating that the regression and dimensional reduction parts in (1) are further decomposed into the two parts according to the two variable groups.

The expression of $f_{GSCA} (W, C, B)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{GSCA}(\textbf{W}, \textbf{C}, \textbf{B})$$\end{document} can be rewritten as follows using orthonormal matrices $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} that satisfy $T^{'} T = T T^{'} = I_{D_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}^{\prime }{} \textbf{T}=\textbf{T}{} \textbf{T}^{\prime } = \textbf{I}_{D_{1}}$$\end{document} and $U^{'} U = U U^{'} = I_{D_{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}^{\prime }{} \textbf{U}=\textbf{U}{} \textbf{U}^{\prime } = \textbf{I}_{D_{2}}$$\end{document} , respectively:

(11)

\begin{matrix} f_{GSCA} (W, C, B) = & | | Z_{1} - Z_{1} W_{1} T^{'} T C_{1} {| |}^{2} + | | Z_{2} - Z_{2} W_{2} U^{'} U C_{2} {| |}^{2} \\ + | | Z_{1} W_{1} T^{'} - (Z_{1} W_{1} T^{'} T {\tilde{B}}_{1} T^{'} + Z_{2} W_{2} U^{'} U B_{2} T^{'}) {| |}^{2} \\ + | | Z_{2} W_{2} U^{'} - (Z_{1} W_{1} T^{'} T B_{1} U^{'} + Z_{2} W_{2} U^{'} U {\tilde{B}}_{2} U^{'}) {| |}^{2} . \end{matrix}

This indicates that the parameter matrices in GSCA are rotated as

(12)

\begin{matrix} W_{1} \to W_{1} T^{'} & , & W_{2} \to W_{2} U^{'} \\ C_{1} \to T C_{1} & , & C_{2} \to U C_{2} \\ B_{1} \to T B_{1} U^{'} & , & B_{2} \to U B_{2} T^{'} \\ {\tilde{B}}_{1} \to T {\tilde{B}}_{1} T^{'} & , & {\tilde{B}}_{2} \to U {\tilde{B}}_{2} U^{'} \end{matrix}

without loss optimality in terms of the minimization of $f_{GSCA} (W, C, B)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{GSCA}(\textbf{W}, \textbf{C}, \textbf{B})$$\end{document} . Following the aforementioned result, the parameter matrices in GSCA can be freely rotated by $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} . EGSCA exploits the identifiability and rotates the parameter matrices as described below.

2.2 Exploratory GSCA

Using the indeterminacy of orthogonal rotation shown above, this article proposes a new GSCA procedure called EGSCA, which explores the correspondence between observed variables and components as well as between components by simplifying the parameter matrices using orthogonal rotation.

As stated above, the diagonal elements of ${\tilde{B}}_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_1$$\end{document} and ${\tilde{B}}_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_2$$\end{document} are constrained to be zero to avoid a trivial solution, ${\tilde{B}}_{1} = I_{D_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_1 = \textbf{I}_{D_1}$$\end{document} and ${\tilde{B}}_{2} = I_{D_{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_2 = \textbf{I}_{D_2}$$\end{document} , which means the components are regressed on themselves. It would be impossible to keep the constraint after rotating ${\tilde{B}}_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_{1}$$\end{document} and ${\tilde{B}}_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_2$$\end{document} as defined in (12). We therefore impose another constraint that ${\tilde{B}}_{1} =_{D_{1}} O_{D_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_{1} = {}_{D_1}{} \textbf{O}_{D_1}$$\end{document} and ${\tilde{B}}_{2} =_{D_{2}} O_{D_{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_2 = {}_{D_2}{} \textbf{O}_{D_2}$$\end{document} where $_{t} O_{u}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{t}{} \textbf{O}_{u}$$\end{document} denotes $t \times u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\times u$$\end{document} matrix filled with zeros. The constraint indicates that the components formed by the same variable group are not connected with any paths. EGSCA thus seeks simple relationships between observed variables and components, parameterized as $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} , $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} , $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} , and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} , and connection between components formed by different variable groups noted as $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} .

It is important to mention that the component score is not standardized after rotation, and we call this issue the standardization issue hereafter. Namely, $diag (T Γ_{1}^{'} Γ_{1} T^{'})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{diag}(\textbf{T}\mathbf{\Gamma }_1^{\prime }{\varvec{\Gamma }}_1\textbf{T}^{\prime })$$\end{document} and $diag (U Γ_{2}^{'} Γ_{2} U^{'})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{diag}(\textbf{U}\mathbf{\Gamma }_2^{\prime }{\varvec{\Gamma }}_2\textbf{U}^{\prime })$$\end{document} are not identity matrices. The reason why the standardization issue occurs is that the component scores in $Γ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Gamma }}_1$$\end{document} and $Γ_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Gamma }}_2$$\end{document} are correlated, i.e., $Γ_{1}^{'} Γ_{1} \neq I_{D_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Gamma }}_1^{\prime }{\varvec{\Gamma }}_1 \ne \textbf{I}_{D_1}$$\end{document} and $Γ_{2}^{'} Γ_{2} \neq I_{D_{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Gamma }}_2^{\prime }{\varvec{\Gamma }}_2 \ne \textbf{I}_{D_2}$$\end{document} before rotation even though their sum of squares are restricted to be 1 because of (3).

To avoid the issue, we consider to restrict the initial solution for EGSCA using the constraint that

(13)

\begin{matrix} W_{1}^{'} Z_{1}^{'} Z_{1} W_{1} = I_{D_{1}}, W_{2}^{'} Z_{2}^{'} Z_{2} W_{2} = I_{D_{2}} \end{matrix}

which indicates the component scores are standardized and not correlated. It is easily verified that the component scores are standardized even after the rotation when (13). The existing GSCA procedure by Hwang and Takane (Reference Hwang and Takane2004) minimizes (1) subject to (3) not (13). The article therefore proposes a new procedure for the minimization problem, which is detailed in Appendix A. Though the algorithm in Appendix A produces a different solution compared to the original GSCA procedure because they use the different constraints, the solutions by the former algorithm makes sense as a GSCA’s solution because the constraint (13) is a special case of (3).

Once the initial solution is obtained, the simplest case of EGSCA is to specify orthonormal matrices $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} that simplify the loading matrices $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} , respectively, and explore the simple relationship between the observed variables and components. This is accomplished using existing procedures for orthogonal rotation, such as Varimax rotation (Kaiser, Reference Kaiser1958), for example. The rotated matrices are expected to possess simple structure, and therefore they help to comprehend which components is matched to each of the observed variables.

There are several options for the specification of $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} . In some cases, it is expected that each of the observed variables corresponds to only one component. A similar assumption is often made in the application of factor-based SEM, which is referred to as an independent cluster model (Marsh et al., Reference Marsh, Lüdtke, Muthén, Asparouhov, Morin, Trautwein and Nagengast2010). For example, the transpose of $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} in (4) is given as a perfect cluster structure (Harris and Kaiser, Reference Harris and Kaiser1964; Kaiser, Reference Kaiser1974; Bernaards and Jennrich, Reference Bernaards and Jennrich2003) where each of the rows contains only one nonzero element and zeros elsewhere. To meet the requirement in EGSCA, $C_{1}^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1^{\prime }$$\end{document} and $C_{2}^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2^{\prime }$$\end{document} should approximate a perfect cluster structure by rotation. The special case of Simplimax rotation (Kiers, Reference Kiers1994) with variable complexity restricted to one is thought to be suitable for this purpose.

The other case considers simplifying $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_{1}$$\end{document} and $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_{2}$$\end{document} to extract how the components are formed by the observed variables as a simple structure. This is accomplished by applying orthogonal rotational procedure to $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_{1}$$\end{document} and $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_{2}$$\end{document} . In some situations, one might consider finding simplified relationship between the components by simplifying $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} or $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} . For example, in the case where $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} is simplified, the following iterative procedure is suitable for the purpose:

1. Initialize $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} .
2. Specify $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} by an orthogonal rotation that simplifies $U B_{1}^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}{} \textbf{B}_1^{\prime }$$\end{document} given the current $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} .
3. Specify $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} by an orthogonal rotation that simplifies $T B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}{} \textbf{B}_1$$\end{document} given the current $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} .
4. Repeat 2 and 3 until the changes in $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} converge.
5. Transform $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} as ${TB}_{1} U^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{TB}_1\textbf{U}^{\prime }$$\end{document} .

2.3. Simultaneous Target Rotation

The EGSCA procedure proposed above aims to simplify a part of the parameter matrices, $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} for instance, and the other matrices are rotated by $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} which simplify $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} . Therefore, the parameter matrices other than $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} might be less simplified after the rotation.

The article further proposes a new orthogonal rotation procedure called simultaneous target rotation, which aims to simplify all the parameter matrices simultaneously. The simultaneous target rotation minimizes the following joint criterion:

(14)

\begin{matrix} F_{ST} (T, U, S) = & α_{1} | | W_{1}^{♯} - W_{1} T^{'} {| |}^{2} + α_{2} | | W_{2}^{♯} - W_{2} U^{'} {| |}^{2} \\ + α_{3} | | C_{1}^{♯} - T C_{1} {| |}^{2} + α_{4} | | C_{2}^{♯} - U C_{2} {| |}^{2} \\ + α_{5} | | B_{1}^{♯} - T B_{1} U^{'} {| |}^{2} + α_{6} | | B_{2}^{♯} - U B_{2} T^{'} | | \end{matrix}

where $W_{1}^{♯} (J_{1} \times D_{1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1^{\sharp }(J_1 \times D_1)$$\end{document} , $W_{2}^{♯} (J_{2} \times D_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2^{\sharp }(J_2 \times D_2)$$\end{document} , $C_{1}^{♯} (D_{1} \times J_{1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1^{\sharp }(D_1 \times J_1)$$\end{document} , $C_{2}^{♯} (D_{2} \times J_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2^{\sharp }(D_2 \times J_2)$$\end{document} , $B_{1}^{♯} (D_{1} \times D_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1^{\sharp }(D_1 \times D_2)$$\end{document} , and $B_{2}^{♯} (D_{2} \times D_{1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2^{\sharp }(D_2 \times D_1)$$\end{document} are the target matrices of $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} , $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} , $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} , $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} , $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} , and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} , respectively. $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} is minimized over the rotation matrices $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} , and the set of target matrices $S = {W_{1}^{♯}, W_{2}^{♯}, C_{1}^{♯}, C_{2}^{♯}, B_{1}^{♯}, B_{2}^{♯}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}} = \{ \textbf{W}_1^{\sharp }, \textbf{W}_2^{\sharp }, \textbf{C}_1^{\sharp }, \textbf{C}_2^{\sharp }, \textbf{B}_1^{\sharp }, \textbf{B}_2^{\sharp } \}$$\end{document} . $α_{1}, \dots, α_{6}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1,\ldots ,\alpha _6$$\end{document} denote positive constants, and we set them as

(15)

\begin{matrix} α_{1} = α_{3} = \frac{1}{J_{1} D_{1}}, α_{2} = α_{4} = \frac{1}{J_{2} D_{2}}, α_{5} = α_{6} = \frac{1}{D_{1} D_{2}} \end{matrix}

to normalize the dimensional difference of the six parameter matrices, following Adachi (Reference Adachi2009, Reference Adachi2013), and we call it size-normalized weight. A similar weighting manner is considered in (Kiers, Reference Kiers1998), referred to as “natural weight.” Other choice of the constants are considered in the next subsection. In order to avoid the trivial solution $W_{1}^{♯} = W_{1} T^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1^{\sharp } = \textbf{W}_1\textbf{T}^{\prime }$$\end{document} , for instance, $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} is minimized under some suitable constraint on the target matrices. Here, we consider the constraint that $W_{1}^{♯}, W_{2}^{♯}, C_{1}^{♯'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1^{\sharp }, \textbf{W}_2^{\sharp }, \textbf{C}_1^{\sharp \prime }$$\end{document} , and $C_{2}^{♯'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2^{\sharp \prime }$$\end{document} have only one nonzero entry in each row. In other words, the matrices are assumed to have perfect cluster structure. Note that the constraint is transposed in $C_{1}^{♯}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1^{\sharp }$$\end{document} and $C_{2}^{♯}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2^{\sharp }$$\end{document} ; they are constrained to have only one nonzero entry in each of its columns because the matrices have more columns than rows. In this respect, the minimization of the first to fourth terms of $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} is equivalent to Simplimax (Kiers, 1999) with complexity one for each variable. The constraint is considered to be reasonable; in that it is often assumed that each observed variable is associated with a single latent variable in the common practice of SEM including confirmatory factor analysis model and multiple indicators and multiple cases (MIMIC) model (Wang and Wang, Reference Wang and Wang2019; Marsh et al., Reference Marsh, Morin, Parker and Kaur2014). Further, the elements of $B_{1}^{♯}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1^{\sharp }$$\end{document} and $B_{2}^{♯}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2^{\sharp }$$\end{document} are constrained to be larger than $τ_{1} (> 0)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1 (> 0)$$\end{document} and $τ_{2} (> 0)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _2 (> 0)$$\end{document} in absolute sense, respectively, or equal to zero. The constraint serves to simplify the relationship between the latent variables. The choice of $τ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1$$\end{document} and $τ_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _2$$\end{document} is detailed in the following subsection.

A similar minimization problem was treated in Adachi (Reference Adachi2009, Reference Adachi2013), as a generalization of Procrustes analysis (Gower et al., Reference Gower and Dijksterhuis2004). It is called generalized Procrustes analysis (GPA) which aims to rotate the parameter matrices obtained by principal component analysis applied to multiple data matrices. Let $X_{k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{X}_k$$\end{document} be an $N \times J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \times J$$\end{document} data matrix and $G_{k} (N \times m)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{G}_k (N \times m)$$\end{document} and $H_{k} (J \times m)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{H}_k (J \times m)$$\end{document} are the matrices of component scores and loadings, respectively, for $k = 1, \dots, K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = 1,\ldots ,K$$\end{document} . GPA obliquely rotates $G_{k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{G}_k$$\end{document} and $H_{k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{H}_k$$\end{document} for all ks by minimizing

(16)

\begin{matrix} F_{GPA} (S_{1}, \dots, S_{K}, G^{♯}, H^{♯}) = \frac{1}{Nm} \sum_{k = 1}^{K} | | G^{♯} - G_{k} S_{k} {| |}^{2} + \frac{1}{Jm} \sum_{k = 1}^{K} | | H^{♯'} - S_{k}^{- 1} H_{k}^{'} {| |}^{2} \end{matrix}

subject to some suitable constraints. Apparently, GPA is different from the minimization of (14), in that both row and column spaces of $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} are rotated in the rotation problem considered in this article. Namely, the minimization of $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} is equivalent to the orthogonal GPA where $S_{k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{S}_k$$\end{document} is an orthonormal matrix when we set $α_{5} = α_{6} = 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _5 = \alpha _6 = 0$$\end{document} , $W_{1}^{♯} = W_{2}^{♯}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1^{\sharp } = \textbf{W}_2^{\sharp }$$\end{document} , and $C_{1}^{♯} = C_{2}^{♯}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1^{\sharp } = \textbf{C}_2^{\sharp }$$\end{document} in the simultaneous target rotation. The similar rotational problem is referred as double Procrustes problem (Goodall, Reference Goodall1991; Gower et al., Reference Gower and Dijksterhuis2004), which rotates row and column spaces simultaneously to approximate a target matrix. To the best of the author’s knowledge, the minimization of the joint criterion of double Procrustes problem and orthogonal GPA has not been considered in the context of simplification of multiple parameter matrices.

Kiers (Reference Kiers1998) considered the simultaneous simplification of multiple parameter matrices in three-mode principal component analysis (PCA). In three-mode PCA, the three component matrices $A (N \times P)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{A} (N\times P)$$\end{document} , $B (J \times Q)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}(J\times Q)$$\end{document} , $C (K \times R)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}(K\times R)$$\end{document} , and $P \times Q \times R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\times Q\times R$$\end{document} three-way core array $\underset{̲}{G} = {g_{pqr}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\textbf{G}} = \{ g_{pqr} \}$$\end{document} are transformed as $A T_{A}^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{A}{} \textbf{T}_\textbf{A}^{\prime }$$\end{document} , $B T_{B}^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}\textbf{T}_\textbf{B}^{\prime }$$\end{document} , $C T_{C}^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}{} \textbf{T}_\textbf{C}^{\prime }$$\end{document} , and $\sum_{i}^{P} \sum_{j}^{Q} \sum_{k}^{R} t_{ip}^{A} t_{jq}^{B} t_{kr}^{C} g_{pqr}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i}^{P}\sum _{j}^{Q}\sum _{k}^{R}t^\textbf{A}_{ip}t^\textbf{B}_{jq}t^\textbf{C}_{kr}g_{pqr}$$\end{document} , respectively, using $T_{A} = {t_{p_{i} p_{j}}^{A}} (P \times P)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}_\textbf{A} = \{ t^\textbf{A}_{p_i p_j} \}(P\times P)$$\end{document} , $T_{B} = {t_{q_{i} q_{j}}^{B}} (Q \times Q)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}_\textbf{B} = \{ t^\textbf{B}_{q_i q_j} \}(Q\times Q)$$\end{document} , and $T_{C} = {t_{r_{i} r_{j}}^{C}} (R \times R)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}_\textbf{C} = \{ t^\textbf{C}_{r_i r_j} \}(R\times R)$$\end{document} orthonormal matrices. In simplifying the three parameter matrices and the core array, the goal is to optimize the joint criterion

(17)

\begin{matrix} F_{JO} = α_{A} f_{OR} (A T_{A}^{'}, γ_{A}) + α_{B} f_{OR} (B T_{B}^{'}, γ_{B}) + α_{C} f_{OR} (A T_{C}^{'}, γ_{C}) + \sum_{l}^{3} α_{G_{l}} f_{OR} ({\tilde{G}}_{l}, γ_{l}) \end{matrix}

over $T_{A}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}_\textbf{A}$$\end{document} , $T_{B}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}_\textbf{B}$$\end{document} , and $T_{C}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}_\textbf{C}$$\end{document} , where $f_{OR} (∙, γ)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{OR}(\bullet ,\gamma )$$\end{document} denotes the orthomax value of a matrix in its first argument with the parameter $γ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} , and ${\tilde{G}}_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{G}}_1$$\end{document} , ${\tilde{G}}_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{G}}_2$$\end{document} , and ${\tilde{G}}_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{G}}_3$$\end{document} are the matrices whose columns are vectorized horizontal, lateral, and frontal slices of the transformed core array. $α_{A}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _\textbf{A}$$\end{document} , $α_{B}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _\textbf{B}$$\end{document} , $α_{C}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _\textbf{C}$$\end{document} , and $α_{G_{l}} (l = 1, 2, 3)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{\textbf{G}_l}\ (l = 1,2,3)$$\end{document} are the positive integers for weighting. Although the objectives of the proposed rotational procedure and Kiers’ algorithm (Reference Kiers1998) are the same, which is to simplify multiple matrices simultaneously, the latter cannot be applied to the rotational problem in GSCA discussed in this article. This is because the rotational problem in GSCA involves doubly rotating the rows and column vectors of the parameter matrices, specifically $T B_{1} U^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}\textbf{B}_1\textbf{U}^{\prime }$$\end{document} and $U B_{1} T^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}{} \textbf{B}_1\textbf{T}^{\prime }$$\end{document} . Further, $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} , $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} , $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} , and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} share the same rotational matrix $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} , while $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} , $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} , $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} share $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} , and this relationship is more complicated than in three-mode PCA. In other words, (14) is not reduced to (17).

For minimizing $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} in (14), $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} , $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} , and $S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {S}}}$$\end{document} are jointly updated given their random initial values, keeping the other matrices fixed until the decrement of the function value converges. To this end, we first rewrite (14) as

(18)

\begin{matrix} F_{ST} (T, U, S) = - 2 α_{1} tr W_{1}^{♯'} W_{1} T^{'} - 2 α_{3} tr C_{1}^{♯'} T C_{1} - 2 α_{5} tr B_{1}^{♯'} T B_{1} U^{'} - 2 α_{6} tr B_{2}^{♯'} U B_{2} T^{'} + c_{T} \end{matrix}

where $c_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\textbf{T}$$\end{document} is the constant that is irrelevant to $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} . The minimization of $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} over $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} is thus equivalent to the maximization of ${\tilde{F}}_{T} (T) = tr 2 (α_{1} W_{1}^{♯'} W_{1} + α_{3} C_{1}^{♯} C_{1}^{'} + α_{5} B_{1}^{♯} U B_{1}^{'} + α_{6} B_{2}^{♯'} U B_{2}) T^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{F}}_\textbf{T}(\textbf{T}) = \textrm{tr}2(\alpha _1\textbf{W}_1^{\sharp \prime }{} \textbf{W}_1 + \alpha _3\textbf{C}_1^{\sharp }{} \textbf{C}_1^{\prime } + \alpha _5\textbf{B}_1^{\sharp }{} \textbf{U}{} \textbf{B}_1^{\prime } + \alpha _6\textbf{B}_2^{\sharp \prime }{} \textbf{U}{} \textbf{B}_2)\textbf{T}^{\prime }$$\end{document} . ${\tilde{F}}_{T} (T)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{F}}_\textbf{T}(\textbf{T})$$\end{document} is maximized as ${\tilde{F}}_{T} (T) = tr K_{T} Λ_{T} L_{T}^{'} \leq tr Λ_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{F}}_\textbf{T}(\textbf{T}) = \textrm{tr}\textbf{K}_\textbf{T}{\varvec{\Lambda }}_\textbf{T}{} \textbf{L}_\textbf{T}^{\prime } \le \textrm{tr}\mathbf{\Lambda }_\textbf{T}$$\end{document} and the equality holds when

(19)

\begin{matrix} T = K_{T} L_{T}^{'} \end{matrix}

with the singular value decomposition of the $D_{1} \times D_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_1 \times D_1$$\end{document} matrix

(20)

\begin{matrix} 2 (α_{1} W_{1}^{♯'} W_{1} + α_{3} C_{1}^{♯} C_{1}^{'} + α_{5} B_{1}^{♯} U B_{1}^{'} + α_{6} B_{2}^{♯'} U B_{2}) = K_{T} Λ_{T} L_{T}^{'} \end{matrix}

where $K_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{K}_\textbf{T}$$\end{document} and $L_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_\textbf{T}$$\end{document} are the matrices of the left and right singular vectors of the left side of (20), and $Λ_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Lambda }}_\textbf{T}$$\end{document} is the diagonal matrix of the singular values arranged in descending order (Ten Berge, Reference Ten Berge1993). In the same manner, the minimization of $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} over $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} is attained by maximizing ${\tilde{F}}_{U} (U) = tr 2 (α_{2} W_{2}^{♯'} W_{2} + α_{4} C_{2}^{♯} C_{2}^{'} + α_{5} B_{1}^{♯'} T B_{1} + α_{6} B_{2}^{♯} T B_{2}^{'}) U^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{F}}_\textbf{U}(\textbf{U}) = \textrm{tr}2(\alpha _2\textbf{W}_2^{\sharp \prime }\textbf{W}_2 + \alpha _4\textbf{C}_2^{\sharp }{} \textbf{C}_2^{\prime } + \alpha _5\textbf{B}_1^{\sharp \prime }{} \textbf{T}{} \textbf{B}_1 + \alpha _6\textbf{B}_2^{\sharp }\textbf{T}{} \textbf{B}_2^{\prime })\textbf{U}^{\prime }$$\end{document} . It is maximized as ${\tilde{F}}_{U} (U) = tr K_{U} Λ_{U} L_{U}^{'} \leq tr Λ_{U}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{F}}_\textbf{U}(\textbf{U}) = \textrm{tr}\textbf{K}_\textbf{U}{\varvec{\Lambda }}_\textbf{U}{} \textbf{L}_\textbf{U}^{\prime } \le \textrm{tr}{\varvec{\Lambda }}_\textbf{U}$$\end{document} , and the maximizer is given by

(21)

\begin{matrix} U = K_{U} L_{U}^{'} \end{matrix}

using the singular value decomposition of the $D_{2} \times D_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2 \times D_2$$\end{document} matrix

(22)

\begin{matrix} 2 (α_{2} W_{2}^{♯'} W_{2} + α_{4} C_{2}^{♯} C_{2}^{'} + α_{5} B_{1}^{♯'} T B_{1} + α_{6} B_{2}^{♯} T B_{2}^{'}) = K_{U} Λ_{U} L_{U}^{'} \end{matrix}

with $Λ_{U}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Lambda }}_\textbf{U}$$\end{document} being the diagonal matrix of the singular values of the left side in (22) arranged in descending order, and $K_{U}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{K}_\textbf{U}$$\end{document} and $L_{U}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_\textbf{U}$$\end{document} denote the matrices of the corresponding left and right singular vectors, respectively.

Further, $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} is minimized over the set of target matrices $S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}$$\end{document} as follows. The optimal target matrices ${\hat{W}}_{1} = {{\hat{w}}_{j_{1} d_{1}}^{(1)}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\textbf{W}}_1 = \{ {\hat{w}}^{(1)}_{j_1d_1} \}$$\end{document} , ${\hat{W}}_{2} = {{\hat{w}}_{j_{2} d_{2}}^{(2)}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\textbf{W}}_2 = \{ {\hat{w}}^{(2)}_{j_2d_2} \}$$\end{document} , ${\hat{C}}_{1} = {{\hat{c}}_{d_{1} j_{1}}^{(1)}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\textbf{C}}_1 = \{ {\hat{c}}^{(1)}_{d_1j_1} \}$$\end{document} , ${\hat{C}}_{2} = {{\hat{c}}_{d_{2} j_{2}}^{(2)}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\textbf{C}}_2 = \{ {\hat{c}}^{(2)}_{d_2j_2} \}$$\end{document} , ${\hat{B}}_{1} = {{\hat{b}}_{d_{1} d_{2}}^{(1)}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\textbf{B}}_1 = \{ {\hat{b}}^{(1)}_{d_1d_2} \}$$\end{document} , and ${\hat{B}}_{2} = {{\hat{b}}_{d_{2} d_{1}}^{(2)}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\textbf{B}}_2 = \{ {\hat{b}}^{(2)}_{d_2d_1} \}$$\end{document} , with $j_{1} = 1, \dots, J_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_1 = 1,\ldots , J_1$$\end{document} , $j_{2} = 1, \dots, J_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_2 = 1,\ldots , J_2$$\end{document} , $d_{1} = 1, \dots, D_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_1 = 1,\ldots , D_1$$\end{document} , and $d_{2} = 1, \dots, D_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_2 = 1,\ldots , D_2$$\end{document} are given by

(23)

\begin{matrix} {\hat{w}}_{j_{1} d_{1}}^{(1)} = (\begin{matrix} {[W_{1} T^{'}]}_{j_{1} d_{1}} & (d_{1} = \underset{d}{arg max} | {[W_{1} T^{'}]}_{j_{1}, d} |) \\ 0 & (o t h e r w i s e) \end{matrix}), \\ {\hat{w}}_{j_{2} d_{2}}^{(2)} = (\begin{matrix} {[W_{2} U^{'}]}_{j_{2} d_{2}} & (d_{2} = \underset{d}{arg max} | {[W_{2} U^{'}]}_{j_{2}, d} |) \\ 0 & (o t h e r w i s e) \end{matrix}) \\ {\hat{c}}_{d_{1} j_{1}}^{(1)} = (\begin{matrix} {[T C_{1}]}_{d_{1} j_{1}} & (d_{1} = \underset{d}{arg max} | {[T C_{1}]}_{d, j_{1}} |) \\ 0 & (o t h e r w i s e) \end{matrix}), \\ {\hat{c}}_{d_{2} j_{2}}^{(2)} = (\begin{matrix} {[U C_{2}]}_{d_{2} j_{2}} & (d_{2} = \underset{d}{arg max} | {[U C_{2}]}_{d, j_{2}} |) \\ 0 & (o t h e r w i s e) \end{matrix}) \\ {\hat{b}}_{d_{1} d_{2}}^{(1)} = (\begin{matrix} {[T B_{1} U^{'}]}_{d_{1} d_{2}} & (| {[T B_{1} U^{'}]}_{d_{1} d_{2}} | > τ_{1}) \\ 0 & (o t h e r w i s e) \end{matrix}), \\ {\hat{b}}_{d_{2} d_{1}}^{(2)} = (\begin{matrix} {[U B_{2} T^{'}]}_{d_{2} d_{1}} & (| {[U B_{2} T^{'}]}_{d_{2} d_{1}} | > τ_{2}) \\ 0 & (o t h e r w i s e) \end{matrix}) \end{matrix}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&{\hat{w}}^{(1)}_{j_1 d_1} = {\left\{ \begin{array}{ll} [\textbf{W}_1\textbf{T}^{\prime }]_{j_1 d_1} &{}(d_1 = \mathop {\text {arg max}}\limits _{d}|[\textbf{W}_1\textbf{T}^{\prime }]_{j_1,d}|)\\ 0 &{}(otherwise) \end{array}\right. },\nonumber \\&{\hat{w}}^{(2)}_{j_2 d_2} = {\left\{ \begin{array}{ll} [\textbf{W}_2\textbf{U}^{\prime }]_{j_2 d_2} &{}(d_2 = \mathop {\text {arg max}}\limits _{d}|[\textbf{W}_2\textbf{U}^{\prime }]_{j_2,d}|)\\ 0 &{}(otherwise) \end{array}\right. }\nonumber \\&{\hat{c}}^{(1)}_{d_1 j_1} = {\left\{ \begin{array}{ll} [\textbf{T}{} \textbf{C}_1]_{d_1 j_1} &{}(d_1 = \mathop {\text {arg max}}\limits _{d}|[\textbf{T}{} \textbf{C}_1]_{d,j_1}|)\\ 0 &{}(otherwise) \end{array}\right. },\nonumber \\&{\hat{c}}^{(2)}_{d_2 j_2} = {\left\{ \begin{array}{ll} [\textbf{U}{} \textbf{C}_2]_{d_2 j_2} &{}(d_2 = \mathop {\text {arg max}}\limits _{d}|[\textbf{U}{} \textbf{C}_2]_{d,j_2}|)\\ 0 &{}(otherwise) \end{array}\right. } \nonumber \\&{\hat{b}}^{(1)}_{d_1 d_2} = {\left\{ \begin{array}{ll} [\textbf{T}{} \textbf{B}_1\textbf{U}^{\prime }]_{d_1 d_2} &{}(|[\textbf{T}{} \textbf{B}_1\textbf{U}^{\prime }]_{d_1 d_2}|> \tau _1)\\ 0 &{}(otherwise) \end{array}\right. },\nonumber \\&{\hat{b}}^{(2)}_{d_2 d_1} = {\left\{ \begin{array}{ll} [\textbf{U}{} \textbf{B}_2\textbf{T}^{\prime }]_{d_2 d_1} &{}(|[\textbf{U}{} \textbf{B}_2\textbf{T}^{\prime }]_{d_2 d_1}| > \tau _2)\\ 0 &{}(otherwise) \end{array}\right. } \end{aligned}$$\end{document}

where ${[\cdot]}_{j, k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\cdot ]_{j,k}$$\end{document} denotes the (j, k)-th element of a matrix in the brackets. To clarify, the optimal target matrices for the weight and loading matrices are obtained by rotating those matrices using the current estimates of $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} . In addition, any elements in the rows of $W_{1} T^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1\textbf{T}^{\prime }$$\end{document} and $W_{2} U^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2\textbf{U}^{\prime }$$\end{document} or columns of $T C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}{} \textbf{C}_1$$\end{document} and $U C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}{} \textbf{C}_2$$\end{document} that are not the highest in absolute value should be replaced with zeros. The optimal target for $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} is simply obtained by $T B_{1} U^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}{} \textbf{B}_1\textbf{U}^{\prime }$$\end{document} and $U B_{2} T^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}{} \textbf{B}_2\textbf{T}^{\prime }$$\end{document} , respectively, but with all the elements less than $τ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1$$\end{document} or $τ_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _2$$\end{document} in absolute sense set to zero.

The discussion above leads to the following iterative algorithm that finds the minimizer of $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} :

1. Initialize $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} and $S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}$$\end{document} .
2. Update $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} by (19) with $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} and $S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}$$\end{document} kept fixed by their current values.
3. Update $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} by (21) with $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}$$\end{document} kept fixed by their current values.
4. Update each entry of $S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}$$\end{document} by (23) with $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} kept fixed by their current values.
5. Stop the algorithm if the decrement of the function value is less than $ϵ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} , otherwise go back to Step 2.

The function value of $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} is guaranteed to decrease at Steps 2 to 4, which implies that the algorithm reaches the minimum after sufficient iterations. Note that the attained minimum may not be a global minimum, and thus the issue of a local minimum is of concern. To avoid the issue, the algorithm starts from $M_{ST} (> 0)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{ST}(> 0)$$\end{document} random initial values and the solution that minimizes $F_{ST} (T, U, S)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{ST}(\textbf{T}, \textbf{U}, {\mathbb {S}})$$\end{document} the most within $M_{ST}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{ST}$$\end{document} solutions is accepted as the final solution. Also, we used $ϵ = 1.0 \times 10^{- 6}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon = 1.0\times 10^{-6}$$\end{document} and $M_{ST} = 100$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{ST} = 100$$\end{document} hereafter.

2.4. Choice of the Weight Constants α₁…α₆ and the Thresholding Constants τ₁ and τ₂

The proposed rotational procedure uses $α_{1}, \dots, α_{6}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1,\ldots ,\alpha _6$$\end{document} as the weight constants that control the relative importance of the six terms of (14) in simplifying the six parameter matrices. As noted above, we used the size-normalized weight as (15), but another choice of the weight constants is possible. Interestingly, the resulting simplicity does not vary significantly, even with another choice of weights, except for the case with some of the weights equal to zero, as demonstrated below.

The conventional GSCA procedure was applied to the drivers’ dataset by Yang et al. (Reference Yang, Chen, Wu, Easa and Zheng2020), which will be detailed in the fourth section, and the simultaneous target rotation was applied to the estimated parameter matrices. In applying the rotation, we used several patterns of the weight constants, shown in Table 1 with short descriptions. Note that $J_{1} = 8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_1 = 8$$\end{document} and $J_{2} = 23$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_2 = 23$$\end{document} for the number of observed variables, and $D_{1} = D_{2} = 3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_1 = D_2 = 3$$\end{document} for the number of components, and $B_{2} =_{3} O_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2 = \ _{3}\textbf{O}_3$$\end{document} was fixed based on the nature of the dataset. The simplicity of the unrotated and rotated matrices were evaluated by the LS index (Lorenzo-Seva, Reference Lorenzo-Seva2003).

The simultaneous rotation significantly simplifies the five parameter matrices, especially for the loading matrices $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} , which are often referred to for specifying the correspondence of observed and latent variables. The attained values of the LS index do not differ in the first five patterns of the weight values, which indicates that the performance of the proposed rotational procedure is stable when different choices of weights are used. This result suggests that it is important whether the parameter matrix is rotated or not, while the relative magnitude of the weights is not.

Nevertheless, for some of the last three patterns, where only a part of the parameter matrices was rotated, the simplicity of the non-rotated matrices could be much higher than before the rotation. For instance, when only $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} and $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} were rotated, the simplicity of $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} is lower than the one attained by the patterns where all matrices were rotated. Further, in the case when only $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} was rotated, the rotated $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} was simplest in all patterns, but the other matrices are less simple than the other patterns. Importantly, the simultaneous rotation of all parameter matrices does not worsen the attained simplicity of $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} or is even better in some cases than in the case only the loading matrices are rotated. The same tendency is observed in the following simulation study, indicating the above observation is not limited to the dataset treated here. The article, therefore, recommends rotating all parameter matrices with size-normalized or identity weight.

Further, the proposed rotational procedure uses $τ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1$$\end{document} and $τ_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _2$$\end{document} to specify the target matrices of $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} , respectively. Table 2 shows the resulting simplicity of the parameter matrices attained by the simultaneous target rotation with $τ_{1} = τ_{2} = τ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1 = \tau _2 = \tau $$\end{document} and the size-normalized weight using the drivers’ dataset. The best LS index value is found in $τ = 0.2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau = 0.2$$\end{document} , while the matrices rotated by other $τ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} settings are less simple. In the simulation studies and the real data examples in the following sections, we used $τ_{1} = τ_{2} = 0.2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1 = \tau _2 = 0.2$$\end{document} , which works properly for simplification of $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} to the best of our experience. Furthermore, the achieved level of simplicity is not greatly impacted by the selection of $τ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} . This implies that one can try with different values of $τ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , such as $0.1, 0.2, \dots, 1.0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.1, 0.2, \ldots , 1.0$$\end{document} , and opt for the value that results in the highest level of simplicity for all parameter matrices.

Table 1 Values of weight constants and the resulting simplicity attained by the simultaneous target rotation applied to the drivers’ dataset.

Table 2 Values of $τ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} parameter and the resulting simplicity attained by the simultaneous target rotation applied to the drivers’ dataset.

3 Numerical Simulation

This section reports the results of the two numerical simulation studies. The first study aims to evaluate the performance of the proposed procedure in terms of how well EGSCA recovers the true value of the parameter matrices. The second study examines the proximity of the suggested model by the proposed procedure to the true model.

3.1. Study 1: Parameter Recovery by EGSCA

3.1.1. Design

The artificial data matrix $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} that contains N observations and J variables was synthesized in the following manner, as originally used in Hwang and Takane (Reference Hwang and Takane2004). The true values of the parameter matrices, noted as $W_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_T$$\end{document} , $C_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_T$$\end{document} , and $B_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_T$$\end{document} , were fixed as follows:

(24)

\begin{matrix} W_{1 T} = & (\begin{matrix} 0.3 & 0.0 \\ 0.4 & 0.0 \\ 0.5 & 0.0 \\ 0.0 & 0.3 \\ 0.0 & 0.4 \\ 0.0 & 0.5 \end{matrix}), W_{2 T} = (\begin{matrix} 0.3 & 0.0 \\ 0.4 & 0.0 \\ 0.5 & 0.0 \\ 0.6 & 0.0 \\ 0.0 & 0.3 \\ 0.0 & 0.4 \\ 0.0 & 0.5 \\ 0.0 & 0.6 \end{matrix}), W_{T} = (\begin{matrix} W_{1 T} \\ W_{2 T} \end{matrix}), \\ C_{1 T} = & W_{1 T}^{'}, C_{2 T} = W_{2 T}^{'}, C_{T} = (\begin{matrix} C_{1 T} \\ C_{2 T} \end{matrix}) \\ B_{1 T} = & (\begin{matrix} 0.2 & 0.9 \\ - 0.9 & 0.2 \end{matrix}), B_{T} = (\begin{matrix} _{2} O_{2} & B_{1 T} \\ _{2} O_{2} & _{2} O_{2} \end{matrix}) . \end{matrix}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{W}_{1T}= & {} \left[ \begin{array}{ll} 0.3 &{} 0.0 \\ 0.4 &{} 0.0 \\ 0.5 &{} 0.0 \\ 0.0 &{} 0.3 \\ 0.0 &{} 0.4 \\ 0.0 &{} 0.5 \\ \end{array} \right] ,\ \textbf{W}_{2T} = \left[ \begin{array}{ll} 0.3 &{} 0.0 \\ 0.4 &{} 0.0 \\ 0.5 &{} 0.0 \\ 0.6 &{} 0.0 \\ 0.0 &{} 0.3 \\ 0.0 &{} 0.4 \\ 0.0 &{} 0.5 \\ 0.0 &{} 0.6 \\ \end{array} \right] ,\ \textbf{W}_T = \left[ \begin{array}{ll} \textbf{W}_{1T} &{} \ \\ \ &{} \textbf{W}_{2T} \\ \end{array} \right] ,\nonumber \\ \textbf{C}_{1T}= & {} \textbf{W}_{1T}^{\prime },\ \textbf{C}_{2T} = \textbf{W}_{2T}^{\prime },\ \textbf{C}_{T} = \left[ \begin{array}{ll} \textbf{C}_{1T} &{} \ \\ \ {} &{} \textbf{C}_{2T} \end{array} \right] \nonumber \\ \textbf{B}_{1T}= & {} \left[ \begin{array}{ll} 0.2 &{} 0.9 \\ -0.9 &{} 0.2 \\ \end{array} \right] ,\ \textbf{B}_T = \left[ \begin{array}{ll} _{2}{} \textbf{O}_2 &{} \textbf{B}_{1T} \\ _{2}{} \textbf{O}_2 &{} _{2}{} \textbf{O}_{2} \\ \end{array} \right] . \end{aligned}$$\end{document}

They indicate that $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} contains 14 variables and are partitioned to six and eight variables. Besides, the path diagram is assumed as indicating that the first six variables $z_{1}, \dots, z_{6}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}_1,\ldots ,\textbf{z}_6$$\end{document} compose the two components $γ_{1} a n d γ_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\gamma }}_1 and {\varvec{\gamma }}_2$$\end{document} , and the last eight variables compose $η_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_1$$\end{document} and $η_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_2$$\end{document} . The four components are connected by the paths as described in Fig. 2. Note that while the four components are fully connected, the relationship between the components is simple, in that the components $γ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\gamma }}_1$$\end{document} and $η_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_2$$\end{document} , and $γ_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\gamma }}_2$$\end{document} and $η_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_1$$\end{document} are strongly connected while weakly for other components. The error terms and paths of the loading matrix are omitted in the figure for the conciseness of the representation. In addition, the true simple structure to be recovered by EGSCA was rotated by randomly generated orthonormal matrices $T_{T} (2 \times 2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}_{T}(2\times 2)$$\end{document} and $U_{T} (2 \times 2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}_{T}(2\times 2)$$\end{document} ; they are transformed as $W_{1 T} \to W_{1 T} T_{T}^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_{1T} \rightarrow \textbf{W}_{1T}{} \textbf{T}_{T}^{\prime }$$\end{document} , $W_{2 T} \to W_{2 T} U_{T}^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_{2T} \rightarrow \textbf{W}_{2T}{} \textbf{U}_{T}^{\prime }$$\end{document} , $C_{1 T} \to T_{T} C_{1 T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_{1T} \rightarrow \textbf{T}_{T}{} \textbf{C}_{1T}$$\end{document} , $C_{2 T} \to U_{T} C_{2 T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_{2T} \rightarrow \textbf{U}_{T}\textbf{C}_{2T}$$\end{document} , and $B_{1 T} \to T_{T}^{'} B_{1 T} U_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_{1T} \rightarrow \textbf{T}_{T}^{\prime }\textbf{B}_{1T}{} \textbf{U}_{T}$$\end{document} .

Figure 2 Path diagram for numerical simulation.

Let $W_{T}^{†}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_T^{\dagger }$$\end{document} and $B_{T}^{†}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_T^{\dagger }$$\end{document} be the last $D_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2$$\end{document} columns of $W_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_T$$\end{document} and $B_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_T$$\end{document} , respectively. They are the matrices of the weight and path coefficients associated with $η_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_1$$\end{document} and $η_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_2$$\end{document} , which are the components treated as dependent variables in the path diagram in Fig. 2. The covariance matrix of error term

(25)

\begin{matrix} E = Z [I_{J}, W_{T}^{†}] - {ZW}_{T} [C_{T}, B_{T}^{†}] \end{matrix}

noted as $Σ_{E}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}_\textbf{E}$$\end{document} is given as Table 10 in Appendix B, where the covariance between the errors associated with the variables in the same group is 0.1, and 0.3 for those are associated with different components. The errors associated with the component scores do not correlate, and all the errors have a unit variance. This is similar with the setting used in Hwang and Takane (Reference Hwang and Takane2004).

Under the setting above, covariance matrix of $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} noted as $Σ_{Z}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}_\textbf{Z}$$\end{document} is given by

(26)

\begin{matrix} Σ_{Z} = {(Φ Φ^{'})}^{- 1} Φ Σ_{E} Φ^{'} {(Φ Φ^{'})}^{- 1} \end{matrix}

with $Φ = [I_{J}, W_{T}^{†}] - W_{T} [C_{T}, B_{T}^{†}]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Phi }} = [\textbf{I}_J, \textbf{W}_T^{\dagger }] - \textbf{W}_T[\textbf{C}_T, \textbf{B}_T^{\dagger }]$$\end{document} . The square root of $Σ_{Z}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}_\textbf{Z}$$\end{document} noted as $R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{R}$$\end{document} satisfying $Σ_{Z} = R R^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}_\textbf{Z} = \textbf{R}{} \textbf{R}^{\prime }$$\end{document} is given by

(27)

\begin{matrix} R = Γ Δ^{1 / 2} \end{matrix}

using the eigenvalue decomposition of $Σ_{Z}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}_\textbf{Z}$$\end{document} as $Σ_{Z} = Γ Δ Γ^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varvec{\Sigma }}_\textbf{Z} = {\varvec{\Gamma }}{\varvec{\Delta }}{\varvec{\Gamma }}^{\prime } $$\end{document} where $Δ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Delta }}$$\end{document} is the diagonal matrix of eigenvalues whose diagonal elements are arranged in descending order, and $Γ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{\Gamma }$$\end{document} is the matrix of corresponding eigenvectors. Using the matrix $Y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Y}$$\end{document} whose elements are randomly generated from the standard normal distribution, the artificial data matrix $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} is generated as

(28)

\begin{matrix} Z = YR . \end{matrix}

The initial solution for EGSCA was obtained using the modified GSCA algorithm in Appendix A applied to $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} with $D_{1} = D_{2} = 2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_1 = D_2 = 2$$\end{document} . Note that the algorithm was started from 100 initial values, and the final solution was selected as the one that minimizes $f_{GSCA} (W, C, B)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{GSCA}(\textbf{W}, \textbf{C}, \textbf{B})$$\end{document} the most among the 100 solutions. The obtained parameter matrices were rotated by simultaneous target rotation with the following three patterns of weight values, refereed to as “size-normalized,” “only $C$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}$$\end{document} s,” and “only $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} ” in “weights” column in Table 3. The first one is the weights in (15) except for $α_{6} = 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _6=0$$\end{document} indicating that $B_{2} =_{2} O_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2 = {}_2\textbf{O}_2$$\end{document} was not rotated, and $α_{1} = α_{2} = α_{5} = α_{6} = 0, α_{3} = {(J_{1} D_{1})}^{- 1}, α_{4} = {(J_{2} D_{2})}^{- 1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1 = \alpha _2 = \alpha _5 = \alpha _6 = 0, \alpha _3=(J_1D_1)^{-1}, \alpha _4=(J_2D_2)^{-1}$$\end{document} and $α_{1} = α_{2} = α_{3} = α_{4} = α_{6} = 0, α_{5} = {(J_{1} D_{1})}^{- 1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1 = \alpha _2 = \alpha _3 = \alpha _4 = \alpha _6 = 0, \alpha _5=(J_1D_1)^{-1}$$\end{document} for the rest, indicating that only ${C_{1}, C_{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\textbf{C}_1, \textbf{C}_2\}$$\end{document} and $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} were rotated, respectively.

The proximity of the rotated parameter matrices $W_{1} T^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1\textbf{T}^{\prime }$$\end{document} , $W_{2} U^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2\textbf{U}^{\prime }$$\end{document} , $T C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}{} \textbf{C}_1$$\end{document} , $U C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}\textbf{C}_2$$\end{document} , and $T B_{1} U^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}{} \textbf{B}_1\textbf{U}^{\prime }$$\end{document} by $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} were obtained by the simultaneous target rotation, and their true values were compared with the congruence coefficient CC (Tucker, Reference Tucker1951; Lorenzo-Seva and Ten Berge, Reference Lorenzo-Seva and Ten Berge2006). The CC measures the proximity of two matrices $M$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{M}$$\end{document} and $M_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{M}_T$$\end{document} having the same dimensionality $P \times K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P \times K$$\end{document} and is defined as

(29)

\begin{matrix} C C (M, M_{T}) = \frac{\sqrt{vec {(M)}^{'} vec (M_{T})}}{\sqrt{vec {(M)}^{'} vec (M)} \sqrt{vec {(M_{T})}^{'} vec (M_{T})}} \end{matrix}

with $vec (M)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {vec}(\textbf{M})$$\end{document} being the PK-dimensional vector obtained by horizontally stacking the column vectors of $M$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{M}$$\end{document} . The same measure is used in Hwang and Takane (Reference Hwang and Takane2004) to evaluate the performance of GSCA in parameter recovery. Note that the rows and/or columns of the true parameter matrices were permuted before the computation of CC so as to match them to the those of the rotated parameter matrices. Namely, permutation matrices $P_{1} (2 \times 2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{P}_1(2\times 2)$$\end{document} and $P_{2} (2 \times 2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{P}_2(2\times 2)$$\end{document} were specified by minimizing

(30)

\begin{matrix} | | W_{1 T} P_{1} - W_{1} T^{'} {| |}^{2} \end{matrix}

and

(31)

\begin{matrix} | | W_{2 T} P_{2} - W_{2} U^{'} {| |}^{2}, \end{matrix}

respectively. The rows and/or columns of $W_{1 T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_{1T}$$\end{document} , $W_{2 T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_{2T}$$\end{document} , $C_{1 T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_{1T}$$\end{document} , $C_{2 T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_{2T}$$\end{document} , $B_{1 T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_{1T}$$\end{document} , and $B_{2 T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_{2T}$$\end{document} were permuted as

(32)

\begin{matrix} W_{1 T} \to W_{1 T} P_{1}, W_{2 T} \to W_{2 T} P_{2} & , & C_{1 T} \to P_{1}^{'} C_{1 T}, C_{2 T} \to P_{2}^{'} C_{2 T}, \\ B_{1 T} \to P_{1}^{'} B_{1 T} P_{2} & , & B_{2 T} \to P_{2}^{'} B_{2 T} P_{1} . \end{matrix}

The simulation considered three cases for N, the number of observations in $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} as $N = 20, 50, 100$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N = 20, 50, 100$$\end{document} . For each condition, the data matrix $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} was generated 100 times, and EGSCA was applied to each of the 100 $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} s. The proximity values of the rotated parameter matrices and their true values were evaluated by CC.

3.1.2. Results

Table 3 shows the averaged CC values and their standard deviations in each of the 2 (N) $\times$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} 3 (pattern of the weights) conditions. The EGSCA procedure sufficiently recovered the true values of the parameter matrices in terms of reasonably high CC with small standard deviations. The CC values get higher as the number of observations N increases with smaller standard deviations. The result indicates that the rotated parameter matrices in EGSCA are sufficiently similar to their true values even when the number of observations is relatively small. The result also indicates the validity of orthogonal rotation in GSCA; in that the rotation not only simplifies the parameter matrices, but also serves to recover their true values.

Further, the best weight value for the three pattern weights is size-normalized weight because the parameter recovery and the attained simplicity are the highest on average with smaller standard deviation in all cases. The CC values are comparable in the first two weighting manners, as demonstrated in the previous section, but the higher values are found in size-normalized weight.

Table 3 Average and standard deviation of CC values of estimated parameters attained by EGSCA with different sizes of samples (N) and patterns of weights.

3.2. Study 2: Accuracy of the Suggested Model by EGSCA

3.2.1. Design

The objective of the second study is to verify whether the EGSCA procedure can identify the true model, and the performance in identification is compared to the one by regularized GSCA (RGSCA).

The data were generated in the same way as in the previous study, with the same true parameters in (24). EGSCA was applied to the dataset with size-normalized weight.

As demonstrated in the next section, one can exploit the EGSCA’s result to develop a hypothetical model, followed by the confirmatory analysis by the conventional GSCA procedure. To convert an EGSCA’s result to a GSCA model, we consider certain thresholding of the loading matrices. Namely, consider the thresholding operator $T (∙)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T(\bullet )$$\end{document} that replaces the elements of the matrix in the parenthesis less than the row-average and column-average that the element belongs with 0 in absolute sense, and the rest of the elements are substituted with 1. The thresholded matrices $C_{1}^{*} = T (T C_{1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1^{*} = T(\textbf{T}{} \textbf{C}_1)$$\end{document} and $C_{2}^{*} = T (U C_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2^{*} = T(\textbf{U}\textbf{C}_2)$$\end{document} indicate which observed variables should be connected to each of the components, and $B_{1}^{*} = T (T B_{1} U^{'})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}^{*}_1 = T(\textbf{T}{} \textbf{B}_1\textbf{U}^{\prime })$$\end{document} and $B_{2}^{*} = T (U B_{2} T^{'})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}^{*}_2 = T(\textbf{U}{} \textbf{B}_2\textbf{T}^{\prime })$$\end{document} exhibit the connection between the components, with $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} being specified by the simultaneous target rotation. Referring to the thresholded matrices $C_{1}^{*}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1^{*}$$\end{document} , $C_{2}^{*}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2^{*}$$\end{document} , $B_{1}^{*}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1^{*}$$\end{document} , and $B_{2}^{*}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2^{*}$$\end{document} , one can develop a GSCA model where the observed variables and the components are connected. Note that the thresholding is conducted after the convergence of the simultaneous target rotation, and the thresholded $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} , $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} , $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} , and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} matrices are no longer optimal. Such thresholding is often conducted in the application of exploratory factor analysis (Trendafilov, Reference Trendafilov2014; Trendafilov et al., Reference Trendafilov, Fontanella and Adachi2017), where the rotated loading matrix is thresholded to convert to a model analyzed by confirmatory factor analysis. The evaluation conducted here is analogous to the thresholding in factor analysis. Checking the proximity of the thresholded parameter matrices and their true values would measure the model’s accuracy suggested by the EGSCA’s result.

The present study evaluates the proximity of the suggested model by the EGSCA’s result to the true model in Fig. 2 based on (24). To evaluate the proximity, we compared the rotated parameter matrices and their true values using the following measure:

(33)

\begin{matrix} A C C (M, M_{T}) = \frac{\sqrt{| | T (M) ∙ T (M_{T}) {| |}^{2}}}{PK} \end{matrix}

where $∙$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} is the Hadamard product, and $M_{T}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{M}_T$$\end{document} denotes the true value of $M$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{M}$$\end{document} . Equation (33) evaluates the similarity of the two thresholded matrices, and it is maximum 1 means that the two thresholded matrices are identical. In the study, we therefore computed $A C C (T C_{1}, C_{1 T})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC(\textbf{T}{} \textbf{C}_1, \textbf{C}_{1T})$$\end{document} , $A C C (U C_{2}, C_{2 T})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC(\textbf{U}{} \textbf{C}_2, \textbf{C}_{2T})$$\end{document} , and $A C C (T B_{1} U^{'}, B_{1 T})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC(\textbf{T}{} \textbf{B}_1\textbf{U}^{\prime }, \textbf{B}_{1T})$$\end{document} , and the suggested model based on the rotated parameter matrices was considered to be equivalent to the true model based on the true values when the ACC value is close enough to its maximum.

As a competitor to EGSCA, RGSCA was applied to the artificial dataset. The tuning parameters $λ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1$$\end{document} , $λ_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2$$\end{document} , and $λ_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _3$$\end{document} in (6) were specified by the fivefold cross-validation as recommended by Hwang (Reference Hwang2009). Noting the estimated parameter matrices as $C_{1}^{R}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1^{R}$$\end{document} , $C_{2}^{R}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2^{R}$$\end{document} , and $B_{1}^{R}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1^{R}$$\end{document} , the ACC values $A C C (C_{1}^{R}, C_{1 T})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC(\textbf{C}^{R}_1, \textbf{C}_{1T})$$\end{document} , $A C C (C_{2}^{R}, C_{2 T})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC(\textbf{C}^{R}_2, \textbf{C}_{2T})$$\end{document} , and $A C C (B_{1}^{R}, B_{1 T})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ACC(\textbf{B}^{R}_1, \textbf{B}_{1T})$$\end{document} were also computed to evaluate how accurate the suggested model by RGSCA.

For $N = 20, 50$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N = 20, 50$$\end{document} , and 100, a hundred of data matrices $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} were generated, and EGSCA and RGSCA were applied in parallel to the $Z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} s, followed by the computation of the ACC values.

3.2.2. Results

Table 4 shows the averages and the standard deviations of ACC values for $N = 20, 50$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=20,50$$\end{document} , and 100 attained by EGSCA and RGSCA. The higher ACC values with smaller standard deviations in EGSCA than RGSCA indicate that EGSCA correctly specifies the true model by thresholding, while the model suggested by RGSCA is less accurate than the one by EGSCA. This suggests that the EGSCA provides a good guidance for building a GSCA model, which is better than RGSCA.

Table 4 Averages and standard deviations of ACC values for $N = 20, 50, 100$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N = 20, 50, 100$$\end{document} obtained by EGSCA and RGSCA.

4. Real Data Example

The proposed procedure was applied to two real datasets to demonstrate that EGSCA correctly simplifies the relationship between the observed variables and components, and the between components regressional relationship.

4.1. Example 1: Organizational Identification Data

The first example deals with the dataset on organizational identification used in Bergami and Bagozzi (Reference Bergami and Bagozzi2000), where 305 employees in South Korea answered a questionnaire. The questionnaire contains 31 items, and they are classified into the following four categories; organizational identification (six items), organization prestige (eight items), affective commitment based on joy (three items), and affective commitment based on love (four items). The same data were analyzed by Hwang and Takane (Reference Hwang and Takane2004) as a demonstration of GSCA procedure, and their article provides further detail of the dataset.

As mentioned in the second section, EGSCA requires dividing observed variables into two groups to enable orthogonal rotation of the parameter matrices. For the purpose, the example further divided the four categories above into the following two groups: organizational identification as the first group, and the remaining three categories are the second group. These two groups were used as the variable groups in EGSCA, and therefore, we had $305 \times 6$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$305\times 6$$\end{document} matrix $Z_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_1$$\end{document} and $305 \times 15$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$305\times 15$$\end{document} matrix $Z_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_2$$\end{document} . The number of components in each variable group was set to $D_{1} = 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_1 = 1$$\end{document} and $D_{2} = 3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2 = 3$$\end{document} , according to the number of item categories comprising the variable groups. The dimensionality of the parameter matrices are therefore $6 \times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$6\times 1$$\end{document} for $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} , $1 \times 6$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\times 6$$\end{document} for $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} , $15 \times 3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$15\times 3$$\end{document} for $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} , $3 \times 15$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\times 15$$\end{document} for $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} , $1 \times 3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\times 3$$\end{document} for $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} , and $3 \times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\times 1$$\end{document} for $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} . The conventional GSCA procedure was started from 100 initial values, and the final solution was picked from among the 100 solutions.

Because the number of the component for the first variable group is 1, the parameter matrices $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} , $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} , and, $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} cannot be rotated because they have only one row or column. Namely, $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} ’s dimensionality is $1 \times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\times 1$$\end{document} , which means $T = 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}=1$$\end{document} . For specifying the other rotation matrix $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} , the example used the simultaneous target rotation with the weights $α_{2} = α_{4} = {(J_{2} D_{2})}^{- 1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _2= \alpha _4 = (J_2 D_2)^{-1}$$\end{document} , $α_{6} = {(D_{1} D_{2})}^{- 1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _6 = (D_1 D_2)^{-1}$$\end{document} , and $α_{1} = α_{3} = α_{5} = 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1 = \alpha _3 = \alpha _5 = 0$$\end{document} . The initial solution was rotated by the simultaneous target rotation also with 100 initial starts.

Tables 5, 6, and 7 show the rotated parameter matrices obtained by the EGSCA. In the tables, the 21 variables are noted as $z_{1}, \dots, z_{21}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}_1,\ldots ,\textbf{z}_{21}$$\end{document} , and $z_{1}, \dots, z_{6}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}_1,\ldots ,\textbf{z}_6$$\end{document} compose $Z_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_1$$\end{document} , whereas $z_{7}, \dots, z_{21}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}_{7},\ldots ,\textbf{z}_{21}$$\end{document} compose $Z_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_2$$\end{document} . Note that the elements of $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} and $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} are multiplied by $\sqrt{305}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{305}$$\end{document} to constrain the variances of the components to be 1, and, instead, the elements in $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} matrices are divided by $\sqrt{305}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{305}$$\end{document} . The estimated $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} and $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} show that the first six variables highly load the first component noted as $γ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\gamma }}_1$$\end{document} , and this indicates that all six variables in $Z_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_1$$\end{document} compose a single component. $γ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\gamma }}_1$$\end{document} is therefore interpreted as organizational identification. The rotated $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} in Table 6 show that the 15 variables in $Z_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}_2$$\end{document} correspond to the three components in their columns noted as $η_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_1$$\end{document} , $η_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_2$$\end{document} , and $η_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_3$$\end{document} . To emphasize the pattern, the elements in $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} that are larger than both the row- and column-wise averages are bolded. The bolded elements clearly show that $z_{7}, \dots, z_{14}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}_7,\ldots ,\textbf{z}_{14}$$\end{document} , $z_{15}, \dots, z_{18}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}_{15},\ldots ,\textbf{z}_{18}$$\end{document} , and $z_{19}, \dots, z_{21}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}_{19},\ldots ,\textbf{z}_{21}$$\end{document} correspond to $η_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_1$$\end{document} , $η_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_2$$\end{document} , and $η_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_3$$\end{document} , respectively. The correspondence pattern is consistent with the sections in the questionnaire; $η_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_1$$\end{document} , $η_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_2$$\end{document} , and $η_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_3$$\end{document} are thought to represent organization prestige, affective commitment based on joy, and affective commitment based on love, respectively.

The connection between the four components is expressed in the elements of $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}$$\end{document} as shown in Table 7. The pattern in the table suggests which components should be connected or not in a path diagram to be examined by GSCA. First, the component $γ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\gamma }}_1$$\end{document} is regressed on by the rest of the components $η_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_1$$\end{document} , $η_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_2$$\end{document} , and $η_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_3$$\end{document} because of the path coefficients in the first row of Table 7 that are large enough than zero. Conversely, $γ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\gamma }}_1$$\end{document} regresses on $η_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_2$$\end{document} because the first column of Table 7 shows the relatively large element in its (3, 1)-th element suggesting that the two components should be connected.

The hypothetical model based on the interpretation of the EGSCA’s result is shown in Fig. 3 as a path diagram. As shown in Fig. 2, the error terms and paths of the loading matrix are omitted. The model shown as the path diagram was fitted to the same dataset using the conventional GSCA using the R package gesca (Hwang et al., Reference Hwang, Kim, Lee and Park2017a, Reference Hwang, Takane and Jungb). The obtained path coefficients between the components are shown in Fig. 4a. The fitness of the model to the dataset dealt in the example was evaluated as follows by AFIT defined as

(34)

\begin{matrix} A F I T = 1 - \frac{d_{0} | | Z [I, W] - {ZW [C, B] | |}^{2}}{d_{1} {| | Z [I, W] | |}^{2}} \end{matrix}

where $d_{0} = N J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_0 = NJ$$\end{document} and $d_{1} = N J - ρ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_1 = NJ - \rho $$\end{document} are the degree of freedom in the null model and the model being examined, respectively, with $ρ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} being the number of free parameters (Hwang et al., Reference Hwang and Takane2014). The resulting AFIT value obtained by the present model in Fig. 3 was 0.547. By contrast, Hwang and Takane (Reference Hwang and Takane2004) proposed a different model of the same dataset in their proposed GSCA procedure, and its path coefficients obtained by GSCA are shown in Fig. 4b. The only difference between the two models is the paths between the four components and their direction as shown in Fig. 4. However, the AFIT value for Hwang & Takane’s (Reference Hwang and Takane2004) model was 0.532, which is inferior to the one suggested by EGSCA. The model in Fig. 4 describes positive feedback between $γ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\gamma }}_1$$\end{document} and $η_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\eta }}_2$$\end{document} , and the feedback was not considered in Hwang and Takane’s (Reference Hwang and Takane2004) model.

The example shown here illustrates that EGSCA serves to extract the simplified relationship between the observed variables and components. Importantly, it is demonstrated that the model suggested by the EGSCA solution is better than that proposed in the previous research in terms of AFIT. This indicates that EGSCA provides beneficial information to GSCA users in creating a better model of a given dataset.

Table 5 Weight ( $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} ) and loading ( $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} ) matrices rotated by simultaneous target rotation.

Table 6 Weight ( $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} ) and loading ( $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} ) matrices rotated by simultaneous target rotation. The elements of $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} larger than row- and column-wise average in absolute sense are bolded.

Table 7 Path coefficient matrix rotated by simultaneous target rotation. Blank cells are the elements equaling to zero.

Figure 3 Path diagram suggested by the result of EGSCA.

Figure 4 Path diagram of a the model suggested by EGSCA and b the one by Hwang and Takane (Reference Hwang and Takane2004).

4.2. Example 2: Drivers’ Data

The second application deals with the dataset obtained and analyzed by Yang et al. (Reference Yang, Chen, Wu, Easa and Zheng2020). The second example differs from the first; in that the former uses two rotation matrices to simplify the parameter matrices.

Three hundred and twenty-four drivers answered the 31 questions about their driving experience. These questions are listed in Tables 8 and 9; see Yang et al. (Reference Yang, Chen, Wu, Easa and Zheng2020) for further detail on the questionnaire. Yang et al. (Reference Yang, Chen, Wu, Easa and Zheng2020) used factor-based SEM to investigate the relationship between the awareness of the driving situation and the factors specific to drivers or roads. They used the factor-based SEM procedure and thus fixed the correspondence between the observed and latent variables based on the questionnaire blocks. On the other hand, the current application treats the correspondence as unknown and uses EGSCA to find the simple relationship using the simultaneous target rotation. In further detail, we divided the 33 observed variables into two groups, eight variables and the others, and assumed that they composed three components respectively, which means $(J_{1}, J_{2}, D_{1}, D_{2}) = (8, 23, 3, 3)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(J_1, J_2, D_1, D_2) = (8, 23, 3, 3)$$\end{document} . The variables in the first group are about the awareness of the driving situation, and those in the other group are about the factors of the drivers and the roads. Further, considering the causal direction between the awareness of the driving situation and drivers’/roads’ factors, we set $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} as a free parameter. On the other hand, $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} was fixed to $_{3} O_{3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ _3\textbf{O}_3$$\end{document} since it represents the effect of the awareness to the factors of drivers and roads, which is unnatural to be assumed. Therefore, the goal of EGSCA is to specify the two rotational matrices $T (3 \times 3)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T} (3\times 3)$$\end{document} and $U (3 \times 3)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}(3\times 3)$$\end{document} that simplify the five parameter matrices, $W_{1} (8 \times 3)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1 (8 \times 3)$$\end{document} , $W_{2} (28 \times 3)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2 (28 \times 3)$$\end{document} , $C_{1} (3 \times 8)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1 (3 \times 8)$$\end{document} , $C_{2} (3 \times 28)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2 (3 \times 28)$$\end{document} , and $B_{1} (3 \times 3)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1 (3 \times 3)$$\end{document} . Note that we used the size-normalized weight for $α_{1}, \dots, α_{5}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1,\ldots ,\alpha _5$$\end{document} .

Tables 8 and 9 show the rotated weight and loading matrices, which represent simple structures. Note that the elements of weight and loading matrices are multiplied and divided by $\sqrt{324}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{324}$$\end{document} , respectively, as in the first example. The three components of the drivers’ and roads’ factors are interpreted as poor road condition, driver’s experience, and positive and energetic mood. The first component is the road-specific factor, while the rest are specific to drivers. The rest of the questions are composed of the following three components, and they are interpreted as the driver’s cognitive ability, distracting elements, and situation awareness. The interpretation is consistent with the one by Yang et al. (Reference Yang, Chen, Wu, Easa and Zheng2020). Interestingly, some variables show connections to multiple components, while most are connected to a single component. For example, “visual processing” seems to corresponds to the first and the third component. In other words, the first and third components share some aspects. Yang et al. (Reference Yang, Chen, Wu, Easa and Zheng2020)’s analysis assumed the independent cluster model where each observed variable is connected to a single latent variable. However, the current application of EGSCA revealed a more complicated relationship between the variables and factors.

The rotated $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} is visualized as a path diagram shown in Fig. 5 where the paths with coefficients less than 0.2 are omitted. The relationship in the figure shows a simple structure, as in the loading matrices, in that over half of the paths can be ignored because of their smaller path coefficients. For instance, poor road condition reduces the drivers’ situation awareness of their driving situation, while the drivers’ experience improves their awareness when they are in a positive and energetic mood. The cognitive performance of drivers is improved by their experience manifested by their age, for example.

The current example also demonstrates how the EGSCA deals with real-world problems. The simultaneous target rotation successfully specified two rotational matrices, simplifying the five parameter matrices simultaneously.

Table 8 Rotated weight matrix $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} and loading matrix $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} of drivers’ data. Elements larger than row- and column-wise average in absolute sense are bolded in the loading matrix.

Table 9 Rotated weight matrix $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} and loading matrix $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} of drivers’ data. Elements larger than row- and column-wise average in absolute sense are bolded in the loading matrix.

Figure 5 Path diagram suggested by rotated $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} of driver’s data. Width of paths are proportional of the corresponding elements in $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} . Solid line denotes a positive coefficient, and dashed line is a negative one. The coefficients less than 0.2 in absolute sense are omitted.

5. Conclusion

GSCA, a component-based SEM method, constructs components as weighted sum scores of multiple variables. An application of GSCA necessitates that the correspondence between the observed variables and components be specified in advance. GSCA verifies the causal relationship between variables and components, and thus GSCA is a CDA procedure. Regarding GSCA, the study first showed that there is an indeterminacy of orthogonal rotation of the parameter matrix in GSCA, by modifying its mathematical formulation. Based on the indeterminacy, the six parameter matrices are permitted to be rotated by two orthonormal matrices $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} . The research further proposed an exploratory variation of GSCA named EGSCA. GSCA requires specifying the relationship between observed variables and components, which is not required in EGSCA; it estimates the correspondence by orthogonal rotation of the parameter matrices. EGSCA exploits the conventional GSCA algorithm to obtain an initial solution of the parameter matrix and then rotates them toward a simple structure. EGSCA therefore exploratorily identifies the correspondence between observed variables and components, as well as the regression relationship between components. In addition, we developed a new rotation algorithm for the parameter matrices, simultaneous target rotation, which is specialized for the purpose of EGSCA. The algorithm defines the simplicity of the parameter matrices in terms of the Procrustes criterion and specifies the rotation matrices $T$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{T}$$\end{document} and $U$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{U}$$\end{document} to minimize the sum of Procrustes criteria of the six parameter matrices. By doing so, simultaneous target rotation aims to simplify all parameter matrices simultaneously.

The numerical simulations investigated whether EGSCA and simultaneous target rotation could reproduce the true values of parameter matrices by applying the proposed procedures to artificial data matrices. The results demonstrated that the proposed method could correctly recover the true values, indicating the rationale of the orthogonal rotation of the parameter matrices in terms of their simplicity and the proximity to their true values after rotation. Further, EGSCA is shown to be better than RGSCA in specifying the true model. Two applications of EGSCA to real datasets were presented. The first application suggested a new causal model of the organizational identification of employees, which is partially different from that in the previous study. Importantly, this new model was confirmed to be superior to the model in the previous study in terms of goodness-of-fit measured by AFIT. This exemplifies that EGSCA can provide useful insights for model exploration, and, at the same time, it suggests the effectiveness of EGSCA in the analysis of real data. In the second demonstration, it was shown how the EGSCA procedure could simplify the parameter matrices using two rotation matrices.

Due to the similarity between ESEM’s rotation and EGSCA, implementing EGSCA’s ideas is expected to improve the ESEM procedure. The measurement model in factor-based SEM is expressed as

(35)

\begin{matrix} z = Λ f + u \end{matrix}

where the $z (J \times 1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}\ (J\times 1)$$\end{document} and $f (D \times 1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{f}\ (D\times 1)$$\end{document} are the vectors of J observed variables and the score vector of D common factors, respectively, and $u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{u}$$\end{document} is the P-dimensional vector of residual (Bartholomew et al., Reference Bartholomew, Knott and Moustaki2011). Note that $z$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{z}$$\end{document} , $f$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{f}$$\end{document} , and $u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{u}$$\end{document} are treated as vectors of random variables. The covariance matrix of the common factor score $V [f] = Φ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V[\textbf{f}] = \mathbf{\Phi }$$\end{document} is not the identity matrix, which means the factors are not orthogonal. This is because the factors are regressed on each other based on the scheme of the structural model defined as

(36)

\begin{matrix} f = B f + ϵ \end{matrix}

where $ϵ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\epsilon }}$$\end{document} denotes the random vector for the residual. The factors are often constrained to have a unit variance and the diagonal elements of $Φ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Phi }}$$\end{document} are restricted to be 1. The rotated factors do not have unit variance in ESEM; the standardization issue occurs in ESEM. According to the ESEM procedure proposed by Asparouhov and Muthén (Reference Asparouhov and Muthén2009), ESEM rotates the factor loading matrix $Λ (P \times D)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Lambda }}(P\times D)$$\end{document} in (35) as $Λ \to Λ S^{- 1'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Lambda }} \rightarrow \mathbf{\Lambda }{} \textbf{S}^{-1\prime }$$\end{document} to maximize the simplicity of $Λ S^{- 1'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{\Lambda }{} \textbf{S}^{-1\prime }$$\end{document} , and it indicates that $f$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{f}$$\end{document} is rotated as $f \to S^{'} f$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{f} \rightarrow \textbf{S}^{\prime }{} \textbf{f}$$\end{document} where $S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{S}$$\end{document} denotes the $D \times D$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\times D$$\end{document} matrix which satisfies $diag (S^{'} S) = I_{D}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{diag}(\textbf{S}^{\prime }{} \textbf{S}) = \textbf{I}_D$$\end{document} . The covariance matrix of $f$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{f}$$\end{document} after the rotation is then expressed as

(37)

\begin{matrix} V [S^{'} f] = S^{'} Φ S \end{matrix}

and its diagonal elements are no longer 1 s because $Φ \neq I_{D}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Phi }} \ne \textbf{I}_D$$\end{document} . Further, the matrix of path coefficients $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}$$\end{document} is accordingly rotated as $S B S^{- 1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{S}{} \textbf{B}{} \textbf{S}^{-1}$$\end{document} , and its diagonal elements are no longer equal to 0, which indicates the self-regression of each of the factors. Moreover, $S$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{S}$$\end{document} is specified to simplify only $Λ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Lambda }}$$\end{document} , while the other parameter matrix $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}$$\end{document} is of interest. However, the proposed EGSCA procedure offers a solution to the aforementioned problems in ESEM, and it is achieved by imposing zero constraints on specific parameter matrices, implementing an orthogonal solution for the initial solution, and optimizing the joint criterion for simplifying all parameter matrices. The standardization issue is ESEM is also solved by splitting the observed variables into two sets and constraining the common factors to be orthogonal within the variable set. These same principles can also be applied to ESEM to address the issues previously mentioned.

It should be noted that the EGSCA does not provide an automatic process in model exploration. That is, one should devise a model to be used as an input of GSCA by referring to the parameter matrices after rotation as illustrated in the fourth section. For the purpose, a threshold would be required to substitute some of the elements in $W$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}$$\end{document} , $C$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}$$\end{document} , and $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}$$\end{document} with 0, which is commonly referred to as thresholding in factor analysis. In the real data analysis in the fourth section, we used the row- and column-wise averages of $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} as their thresholds; the elements less than the two averages were treated as zero. However, how to specify such threshold value still remains an open question, which is a possible direction for future study. Conversely, RGSCA shrinks some elements of the parameter matrix to zero, by means of regularization. Therefore, RGSCA can serve a similar purpose as thresholding, only if the regularization parameters can be set correctly. However, the EGSCA’s performance in model identification performance is better than the one by RGSCA, as empirically shown in the numerical simulation.

To the best of the author’s knowledge, this is the first research that extends GSCA to exploratory purpose by rotation, and thus many points remain to be considered. To conclude the article, we list some possible directions for the future study. First, from the point of statistical inference, how to compute the standard errors after rotation should be established. As mentioned in Hwang and Takane (Reference Hwang and Takane2004), the bootstrapping method can obtain GSCA’s standard errors before rotation. Incorporating bootstrapping and the works by Jennrich (Reference Jennrich1974, Reference Jennrich2007), which proposed a simple formula to compute standard errors after rotation in factor analysis, it would be possible to compute standard errors after rotation in EGSCA. Second, a detailed comparison of other rotational procedures is beneficial for users of EGSCA. The paper proposed the simultaneous rotation procedure, but the conventional rotational procedure is applicable for simplifying a part of the parameter matrices. For example, if one wants to simplify $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} only, it would be reasonable to rotate $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} by Varimax rotation. The simulation study conducted in the article considered the limited situation because its purpose is to evaluate EGSCA’s procedure, and thus an extensive comparison to other rotation procedures is necessary. Third, further application examples on real data should clarify various practical issues, which would be the guideline for further extension of the proposed method. For instance, we used thresholding to convert the rotated EGSCA solution to the GSCA model, but determining the thresholding value is unclear. Using the $L_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_1$$\end{document} penalty, which produces sparse estimates for the parameter matrices, would be a possible extension of EGSCA.

Acknowledgements

The author is deeply grateful to the reviews and the associate editor for their careful reviews and constructive comments for improving the quality of the paper. The author also thanks Professor Henk Kiers at the University of Groningen for his helpful advice.

Funding

This research was supported by JSPS KAKENHI Grant Number 23K16854.

Declarations

Conflict of interest

We have no conflicts of interest to disclose.

Data Availability

R source code for the proposed method can be obtained from the author on request.

Appendix A

In this appendix, we consider the minimization problem

(38)

\begin{matrix} ϕ = & | | Z_{1} - Z_{1} W_{1} C_{1} {| |}^{2} + | | Z_{2} - Z_{2} W_{2} C_{2} {| |}^{2} \\ + | | Z_{1} W_{1} - Z_{2} W_{2} B_{2} {| |}^{2} + | | Z_{2} W_{2} - Z_{1} W_{1} B_{1} {| |}^{2} \end{matrix}

over $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} , $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} , $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} , $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} , $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} , and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} , subject to the constraint

(39)

\begin{matrix} W_{1}^{'} Z_{1}^{'} Z_{1} W_{1} = I_{D_{1}}, W_{2}^{'} Z_{2}^{'} Z_{2} W_{2} = I_{D_{2}} . \end{matrix}

The loss function in (38) is equivalent to the minimization of (10) with ${\tilde{B}}_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_1$$\end{document} and ${\tilde{B}}_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{B}}_2$$\end{document} being the matrices filled with zeros, as assumed in the EGSCA procedure. The component score is constrained to be orthogonal within the variable set in order to avoid the standardization issue after the rotation of the parameter matrices. The estimated parameter matrices are used for an initial solution for the EGSCA procedure. All elements in the parameter matrices are treated as free parameters to be estimated, while some are fixed in the conventional GSCA procedure.

The following iterative algorithm is used for minimizing (38).

1. Initialize $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} , $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} , $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} , $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} , and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} .
2. Update $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} by the one minimizing $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} with other parameter matrices kept fixed.
3. Update $W_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} by the one minimizing $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} with other parameter matrices kept fixed.
4. Update $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} by the ones minimizing $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} with other parameter matrices kept fixed.
5. Update $B_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} and $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} by the ones minimizing $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} with other parameter matrices kept fixed.
6. Terminate the algorithm if the decrement of $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} value is less than $ϵ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} , otherwise go back to Step 2.

The loss function is guaranteed to decrease at Steps 2–5, and the algorithm starts from $M_{ST}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{ST}$$\end{document} initial values to avoid local minimum. $M_{ST} = 100$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{ST} = 100$$\end{document} and $ϵ = 1.0 \times 10^{- 6}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon = 1.0 \times 10^{-6}$$\end{document} were used for all the simulation studies and the applications.

The update formulae for the Steps 2–5 are presented in the following.

First, consider to minimize $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} over $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} subject to (39). $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} is expanded as

(40)

\begin{matrix} ϕ = & - 2 tr Z_{1}^{'} Z_{1} W_{1} C_{1} + tr C_{1}^{'} C_{1} - 2 tr Z_{2}^{'} Z_{2} W_{2} C_{2} + tr C_{2}^{'} C_{2} \\ - 2 tr W_{1}^{'} Z_{1}^{'} Z_{2} W_{2} B_{2} + tr B_{2}^{'} B_{2} - 2 tr W_{2}^{'} Z_{2}^{'} Z_{1} W_{1} B_{1} + tr B_{1}^{'} B_{1} + c \end{matrix}

where c denotes the constant irrelevant to the parameter matrices. (40) indicates that the minimization of $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} over $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} is equivalent to maximize

(41)

\begin{matrix} ϕ_{W_{1}} = & tr Z_{1}^{'} Z_{1} W_{1} C_{1} + W_{1}^{'} Z_{1}^{'} Z_{2} W_{2} B_{2} + tr W_{2}^{'} Z_{2}^{'} Z_{1} W_{1} B_{1} \\ = & tr W_{1}^{'} M_{1} \end{matrix}

where $M_{1} = Z_{1}^{'} Z_{1} C_{1}^{'} + Z_{1}^{'} Z_{2} W_{2} (B_{1}^{'} + B_{2})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{M}_1 = \textbf{Z}_1^{\prime }{} \textbf{Z}_1\textbf{C}_1^{\prime } + \textbf{Z}_1^{\prime }{} \textbf{Z}_2\textbf{W}_2(\textbf{B}_1^{\prime } + \textbf{B}_2)$$\end{document} with the dimensionality of $J_{1} \times D_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_1 \times D_1$$\end{document} . Here, using ${\tilde{W}}_{1} = {(Z_{1}^{'} Z_{1})}^{1 / 2} W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{W}}_1 = (\textbf{Z}_1^{\prime }{} \textbf{Z}_1)^{1/2}{} \textbf{W}_1$$\end{document} , the first constraint in (39) can be rewritten as

(42)

\begin{matrix} {\tilde{W}}_{1}^{'} {\tilde{W}}_{1} = I_{D_{1}} \end{matrix}

and $ϕ_{W_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\textbf{W}_1}$$\end{document} becomes

(43)

\begin{matrix} ϕ_{W_{1}} = tr {\tilde{W}}_{1}^{'} {(Z_{1}^{'} Z)}^{- 1 / 2} M_{1} \end{matrix}

Thus, the minimization of $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} over $W_{1}^{'} Z_{1}^{'} Z_{1} W_{1} = I_{D_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1^{\prime }\textbf{Z}_1^{\prime }{} \textbf{Z}_1\textbf{W}_1 = \textbf{I}_{D_1}$$\end{document} is equivalent to maximizing $ϕ_{W_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\textbf{W}_1}$$\end{document} over the column-orthonormal matrix ${\tilde{W}}_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{W}}_1$$\end{document} . $ϕ_{W_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\textbf{W}_1}$$\end{document} is maximized as

(44)

\begin{matrix} ϕ_{W_{1}} = tr {\tilde{W}}_{1}^{'} K_{1} Λ_{1} L_{1}^{'} \leq tr Λ_{1} \end{matrix}

using the singular value decomposition

(45)

\begin{matrix} {(Z_{1}^{'} Z_{1})}^{- 1 / 2} M_{1} = K_{1} Λ_{1} L_{1}^{'} \end{matrix}

where $K_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{K}_{1}$$\end{document} and $L_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_{1}$$\end{document} are the matrices of the left and right singular vectors of ${(Z_{1}^{'} Z)}^{- 1 / 2} M_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\textbf{Z}_1^{\prime }{} \textbf{Z})^{-1/2}\textbf{M}_1$$\end{document} , respectively, and $Λ_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Lambda }}_{1}$$\end{document} is the diagonal matrix of the singular value arranged in descending order (Ten Berge, Reference Ten Berge1993). The equality in (45) holds when ${\tilde{W}}_{1} = K_{1} L_{1}^{'}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\textbf{W}}_1 = \textbf{K}_{1}{} \textbf{L}_{1}^{\prime }$$\end{document} which leads

(46)

\begin{matrix} W_{1} = {(Z_{1}^{'} Z_{1})}^{- 1 / 2} K_{1} L_{1}^{'} \end{matrix}

as the update formula for $W_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} in Step 2.

The update formula for the subsequent step is similarly given by

(47)

\begin{matrix} W_{2} = {(Z_{2}^{'} Z_{2})}^{- 1 / 2} K_{2} L_{2}^{'} \end{matrix}

where the columns of $K_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{K}_2$$\end{document} and $L_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_2$$\end{document} are the left and right singular vectors of ${(Z_{2}^{'} Z_{2})}^{- 1 / 2} (Z_{2}^{'} Z_{2} C_{2}^{'} + Z_{2}^{'} Z_{1} W_{1} (B_{1} + B_{2}^{'}))$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\textbf{Z}_2^{\prime }{} \textbf{Z}_2)^{-1/2}(\textbf{Z}_2^{\prime }{} \textbf{Z}_2\textbf{C}_2^{\prime } + \textbf{Z}_2^{\prime }{} \textbf{Z}_1\textbf{W}_1(\textbf{B}_1 + \textbf{B}_2^{\prime }))$$\end{document} , respectively.

The $C_{1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} and $C_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} minimizing $ϕ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} is simply obtained by the multivariate regression;

(48)

\begin{matrix} C_{1} = W_{1}^{'} Z_{1}^{'} Z_{1}, C_{2} = W_{2}^{'} Z_{2}^{'} Z_{1} . \end{matrix}

The update formulae in Step 5 are also given by

(49)

\begin{matrix} B_{1} = W_{1}^{'} Z_{1}^{'} Z_{2} W_{2}, B_{2} = W_{2}^{'} Z_{2}^{'} Z_{1} W_{1} . \end{matrix}

The second example in the fourth section fixes $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} as $_{D_{2}} O_{D_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{D_2}{} \textbf{O}_{D_1}$$\end{document} , and it is accomplished by setting $B_{2} =_{D_{2}} O_{D_{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2 = \ _{D_2}{} \textbf{O}_{D_1}$$\end{document} in Step 1, and suppressing the update of $B_{2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_2$$\end{document} in Step 5.

Appendix B

Table 10. Covariance matrix of error terms in numerical simulation.

Footnotes

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

References

Adachi, K.. (2009). Joint Procrustes analysis for simultaneous nonsingular transformation of component score and loading matrices. Psychometrika, 74, 667–683.CrossRef Google Scholar

Adachi, K.. (2013). Generalized joint Procrustes analysis. Computational Statistics, 28, 2449–2464.CrossRef Google Scholar

Alamer, A.. (2022). Exploratory structural equation modeling (ESEM) and bifactor ESEM for construct validation purposes: Guidelines and applied example. Research Methods in Applied Linguistics, 1.CrossRef Google Scholar

Asparouhov, T., Muthén, B.. (2009). Exploratory structural equation modeling. Structural Equation Modeling: a Multidisciplinary Journal, 16, 397–438.CrossRef Google Scholar

Bartholomew, D. J., Knott, M., Moustaki, I.. (2011). Latent variable models and factor analysis: A unified approach, 3New York: Wiley.CrossRef Google Scholar

Bernaards, C. A., Jennrich, R. I.. (2003). Orthomax rotation and perfect simple structure. Psychometrika, 68, 585–588.CrossRef Google Scholar

Bentler, P. M.. (1980). Multivariate analysis with latent variables: Causal modeling. Annual Review of Psychology, 31, 419–456.CrossRef Google Scholar

Bentler, P. M.. (1986). Structural modeling and Psychometrika: An historical perspective on growth and achievements. Psychometrika, 51, 35–51.CrossRef Google Scholar

Bergami, M., Bagozzi, R. P.. (2000). Self-categorization, affective commitment and group self-esteem as distinct aspects of social identity in the organization. British journal of social psychology, 39, 555–577.CrossRef Google Scholar PubMed

Browne, M.. (1972). Orthogonal rotation to a partially specified target. British Journal of Mathematical and Statistical Psychology, 25, 115–120.CrossRef Google Scholar

Browne, M. W.. (1972). Oblique rotation to a partially specified target. British Journal of Mathematical and Statistical Psychology, 25, 207–212.CrossRef Google Scholar

Browne, M. W.. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111–150.CrossRef Google Scholar

Esposito Vinzi, V., Russolillo, G.. (2013). Partial least squares algorithms and methods. Wiley Interdisciplinary Reviews: Computational Statistics, 5, 1–19.CrossRef Google Scholar

Goodall, C.. (1991). Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society: Series B (Methodological), 53, 285–321.CrossRef Google Scholar

Gower, J. C., Dijksterhuis, G. B.. (2004). Procrustes problems. Oxford: OUP.CrossRef Google Scholar

Harris, C. W., Kaiser, H. F.. (1964). Oblique factor analytic solutions by orthogonal transformations. Psychometrika, 29, 347–362.CrossRef Google Scholar

Hwang, H.. (2009). Regularized Generalized Structured Component Analysis. Psychometrika, 74, 517–530.CrossRef Google Scholar

Hwang, H., Cho, G., Jung, K., Falk, C. F., Flake, J. K., Jin, M. J., Lee, S. H.. (2021). An approach to structural equation modeling with both factors and components: Integrated generalized structured component analysis. Psychological Methods, 26, 273.CrossRef Google Scholar PubMed

Hwang, H., Desarbo, W. S., Takane, Y.. (2007). Fuzzy Clusterwise Generalized Structured Component Analysis. Psychometrika, 72, 181–198.CrossRef Google Scholar

Hwang, H., Kim, S., Lee, S. & Park, T. (2017). gesca: Generalized Structured Component Analysis (GSCA). R package version 1.0.4. https://CRAN.R-project.org/package=gesca.Google Scholar

Hwang, H., Takane, Y.. (2004). Generalized structured component analysis. Psychometrika, 69, 81–99.CrossRef Google Scholar

Hwang, H., Takane, Y.. (2014). Generalized structured component analysis: A component-based approach to structural equation modeling. Boca Raton: CRC Press.CrossRef Google Scholar

Hwang, H., Takane, Y., Jung, K.. (2017). Generalized structured component analysis with uniqueness terms for accommodating measurement error. Frontiers in Psychology, 8, 2137.CrossRef Google Scholar PubMed

Jennrich, R. I.. (1974). Simplified formulae for standard errors in maximum-likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 27, 122–131.CrossRef Google Scholar

Jennrich, R. I. (2007). Rotation methods, algorithms, and standard errors. In Factor analysis at 100 (pp. 329–350). Routledge.Google Scholar

Jöreskog, K. G., & Sörbom, D. (1996). LISREL 8: User’s reference guide. Scientific Software International.Google Scholar

Kaiser, H. F.. (1958). The Varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200.CrossRef Google Scholar

Kaiser, H. F.. (1974). An index of factorial simplicity. Psychometrika, 39, 31–36.CrossRef Google Scholar

Kiers, H. A. L.. (1994). Simplimax: Oblique rotation to an optimal target with simple structure. Psychometrika, 59, 567–579.CrossRef Google Scholar

Kiers, H. A. L.. (1998). Joint orthomax rotation of the core and component matrices resulting from three-mode principal components analysis. Journal of Classification, 15, 245–263.CrossRef Google Scholar

Lorenzo-Seva, U.. (2003). A factor simplicity index. Psychometrika, 68, 49–60.CrossRef Google Scholar

Lorenzo-Seva, U., Ten Berge, J. M.. (2006). Tucker’s congruence coefficient as a meaningful index of factor similarity. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, 2, 57.CrossRef Google Scholar

Marsh, H. W., Guo, J., Dicke, T., Parker, P. D., Craven, R. G.. (2020). Confirmatory factor analysis (CFA), exploratory structural equation modeling (ESEM), and set-ESEM: Optimal balance between goodness of fit and parsimony. Multivariate Behavioral Research, 55, 102–119.CrossRef Google Scholar PubMed

Marsh, H. W., Lüdtke, O., Muthén, B., Asparouhov, T., Morin, A. J., Trautwein, U., Nagengast, B.. (2010). A new look at the big five factor structure through exploratory structural equation modeling. Psychological Assessment, 22, 471–491.CrossRef Google Scholar

Marsh, H. W., Morin, A. J. S., Parker, P. D., Kaur, G.. (2014). Exploratory structural equation modeling: An integration of the best features of exploratory and confirmatory factor analysis. Annual Review of Clinical Psychology, 10, 85–110.CrossRef Google Scholar PubMed

McLarnon, M. J.. (2022). Into the heart of darkness: A person-centered exploration of the Dark Triad. Personality and Individual Differences, 186.CrossRef Google Scholar

Mulaik, S. A.. (1986). Factor analysis and Psychometrika: Major developments. Psychometrika, 51, 23–33.CrossRef Google Scholar

Pianta, R. C., Lipscomb, D., Ruzek, E.. (2022). Indirect effects of coaching on pre-K students’ engagement and literacy skill as a function of improved teacher-student interaction. Journal of School Psychology, 91, 65–80.CrossRef Google Scholar PubMed

Poier, S., Nikodemska-Wołowik, A. M., & Suchanek, M. (2022). How higher-order personal values affect the purchase of electricity storage–Evidence from the German photovoltaic market. Journal of Consumer Behaviour.CrossRef Google Scholar

Ten Berge, J. M.. (1993). Least squares optimization in multivariate analysis. Leiden: DSWO Press, Leiden University.Google Scholar

Tenenhaus, M.. (2008). Component-based structural equation modelling. Total Quality Management, 19, 871–886.CrossRef Google Scholar

Tenenhaus, M., Vinzi, V. E., Chatelin, Y. M., Lauro, C.. (2005). PLS path modeling. Computational Statistics & Data Analysis, 48, 159–205.CrossRef Google Scholar

Tucker, L. R. (1951). A method for synthesis of factor analysis studies (Personnel research section report no. 984). Department of the Army.Google Scholar

Trendafilov, N. T.. (2014). From simple structure to sparse components: A review. Computational Statistics, 29, 431–454.CrossRef Google Scholar

Trendafilov, N. T., Fontanella, S., Adachi, K.. (2017). Sparse exploratory factor analysis. Psychometrika, 82, 778–794.CrossRef Google Scholar

Wang, J., Wang, X.. (2019). Structural equation modeling: Applications using Mplus. New York: Wiley.CrossRef Google Scholar

Wold, S., Sjmöstrmöm, M., Eriksson, L.. (2001). PLS-regression: A basic tool of chemometrics. Chemometrics and Intelligent Laboratory Systems, 58, 109–130.CrossRef Google Scholar

Yang, Y., Chen, M., Wu, C., Easa, S. M., Zheng, X.. (2020). Structural equation modeling of drivers’ situation awareness considering road and driver factors. Frontiers in Psychology, 11, 1601.CrossRef Google Scholar PubMed

Figure 1 An example of path diagram for GSCA.

Table 1 Values of weight constants and the resulting simplicity attained by the simultaneous target rotation applied to the drivers’ dataset.

Table 2 Values of τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} parameter and the resulting simplicity attained by the simultaneous target rotation applied to the drivers’ dataset.

Figure 2 Path diagram for numerical simulation.

Table 3 Average and standard deviation of CC values of estimated parameters attained by EGSCA with different sizes of samples (N) and patterns of weights.

Table 4 Averages and standard deviations of ACC values for N=20,50,100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N = 20, 50, 100$$\end{document} obtained by EGSCA and RGSCA.

Table 5 Weight (W1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document}) and loading (C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document}) matrices rotated by simultaneous target rotation.

Table 6 Weight (W2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document}) and loading (C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document}) matrices rotated by simultaneous target rotation. The elements of C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} larger than row- and column-wise average in absolute sense are bolded.

Table 7 Path coefficient matrix rotated by simultaneous target rotation. Blank cells are the elements equaling to zero.

Figure 3 Path diagram suggested by the result of EGSCA.

Figure 4 Path diagram of a the model suggested by EGSCA and b the one by Hwang and Takane (2004).

Table 8 Rotated weight matrix W1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_1$$\end{document} and loading matrix C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_1$$\end{document} of drivers’ data. Elements larger than row- and column-wise average in absolute sense are bolded in the loading matrix.

Table 9 Rotated weight matrix W2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{W}_2$$\end{document} and loading matrix C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{C}_2$$\end{document} of drivers’ data. Elements larger than row- and column-wise average in absolute sense are bolded in the loading matrix.

Figure 5 Path diagram suggested by rotated B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document} of driver’s data. Width of paths are proportional of the corresponding elements in B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{B}_1$$\end{document}. Solid line denotes a positive coefficient, and dashed line is a negative one. The coefficients less than 0.2 in absolute sense are omitted.

Table 10. Covariance matrix of error terms in numerical simulation.

Article contents

Exploratory Procedure for Component-Based Structural Equation Modeling for Simple Structure by Simultaneous Rotation

Abstract

Keywords

Introduction

2. Proposed Method

2.1. Rotational Indeterminacy in GSCA

2.2 Exploratory GSCA

2.3. Simultaneous Target Rotation

2.4. Choice of the Weight Constants α1…α6 and the Thresholding Constants τ1 and τ2

3 Numerical Simulation

3.1. Study 1: Parameter Recovery by EGSCA

3.1.1. Design

3.1.2. Results

3.2. Study 2: Accuracy of the Suggested Model by EGSCA

3.2.1. Design

3.2.2. Results

4. Real Data Example

4.1. Example 1: Organizational Identification Data

4.2. Example 2: Drivers’ Data

5. Conclusion

Acknowledgements

Funding

Declarations

Conflict of interest

Data Availability

Appendix A

Appendix B

Footnotes

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests

2.4. Choice of the Weight Constants α₁…α₆ and the Thresholding Constants τ₁ and τ₂