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Joint Procrustes analysis for simultaneous nonsingular transformation of component score and loading matrices. (English) Zbl 1179.62084

Summary: In component analysis solutions, post-multiplying a component score matrix by a nonsingular matrix can be compensated by applying its inverse to the corresponding loading matrix. To eliminate this indeterminacy on nonsingular transformation, we propose Joint Procrustes Analysis (JPA) in which component score and loading matrices are simultaneously transformed so that the former matrix matches a target score matrix and the latter matches a target loading matrix. The loss function of JPA is a function of the nonsingular transformation matrix and its inverse, and is difficult to minimize straightforwardly. To deal with this difficulty, we reparameterize those matrices by their singular value decomposition, which reduces the minimization to alternately solving quartic equations and performing the existing multivariate procedures. This algorithm is assessed in a simulation study. We further extend JPA for cases where targets are linear functions of unknown parameters. We also discuss how the application of JPA can be extended in different fields.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
62H99 Multivariate analysis
62P15 Applications of statistics to psychology
65C60 Computational problems in statistics (MSC2010)
15A99 Basic linear algebra
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[1] BenzĂ©cri, J.-P. (1992). Correspondence analysis handbook. New York: Marcel Dekker. · Zbl 0766.62034
[2] Browne, M.W. (1967). On oblique Procrustes rotation. Psychometrika, 32, 125–132.
[3] Gower, J.C., & Dijksterhuis, G.B. (2004). Procrustes problems. Oxford: Oxford University Press. · Zbl 1057.62044
[4] Green, B.F. (1952). The orthogonal approximation of an oblique structure in factor analysis. Psychometrika, 17, 429–440. · Zbl 0049.37601
[5] Greenacre, M.J. (1984). Theory and applications of correspondence analysis. London: Academic Press. · Zbl 0555.62005
[6] Hendrickson, A.E., & White, P.O. (1964). PROMAX: A quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 17, 65–70.
[7] Kaiser, H.F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23, 187–200. · Zbl 0095.33603
[8] Kiers, H.A.L. (1990). Majorization as a tool for optimizing a class of matrix functions. Psychometrika, 55, 417–428. · Zbl 0733.62067
[9] Kiers, H.A.L. (1994). SIMPLIMAX: Oblique rotation to an optimal target with simple structure. Psychometrika, 59, 567–579. · Zbl 0925.62228
[10] Kiers, H.A.L. (1998a). Joint orthomax rotation of the core and component matrices resulting from three-mode principal component analysis. Journal of Classification, 15, 245–263. · Zbl 0917.62055
[11] Kiers, H.A.L. (1998b). Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure. Computational Statistics and Data Analysis, 28, 307–324. · Zbl 1042.62559
[12] Kiers, H.A.L., & ten Berge, J.M.F. (1992). Minimization of a class of matrix trace functions by means of refined majorization. Psychometrika, 57, 371–382. · Zbl 0782.62067
[13] Kroonenberg, P.M. (2008). Applied multiway data analysis. Hoboken: Wiley. · Zbl 1160.62002
[14] Kroonenberg, P.M., & de Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 45, 69–97. · Zbl 0431.62035
[15] Lorenzo-Seva, U., & ten Berge, J.M.F. (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, 57–64.
[16] Mori, K. (1990). Nonlinear equations. In Y. Ohno & K. Isoda (Eds.), Handbook of Numerical Computations: A New Version (pp. 589–688). Tokyo: Ohm-Sha (in Japanese).
[17] Mosier, C.I. (1939). Determining a simple structure when loadings for certain tests are known. Psychometrika, 4, 149–162. · JFM 65.0604.03
[18] Mulaik, S.A. (1972). The foundations of factor analysis. New York: McGraw-Hill. · Zbl 1182.62133
[19] Osgood, C.E., Suci, G.J., & Tannenbaum, P.H. (1957). The measurement of meaning. Urbana: University of Illinois Press.
[20] Takane, Y., & Shibayama, T. (1991). Principal component analysis with external information of both subjects and variables. Psychometrika, 56, 97–120. · Zbl 0725.62055
[21] ten Berge, J.M.F. (1993). Least squares optimization in multivariate analysis. Leiden: DSWO Press. · Zbl 0937.62542
[22] ten Berge, J.M.F., & Kiers, H.A.L. (1996). Optimality criteria for principal component analysis and generalizations. British Journal of Mathematical and Statistical Psychology, 49, 335–345. · Zbl 0925.62218
[23] ten Berge, J.M.F., & Nevels, K. (1977). A general solution of Mosier’s oblique procrustes problem. Psychometrika, 42, 593–600. · Zbl 0387.62086
[24] Tucker, L.R. (1951). A method for synthesis of factor analysis studies (Personnel Research Section Report No. 984). Department of the Army, Washington, DC.
[25] Tucker, L.R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.
[26] van de Velden, M., & Kiers, H.A.L. (2003). An application of rotation in correspondence analysis. In H. Yanai, A. Okada, K. Shigemasu, Y. Kano, & J.J. Meulman (Eds.), New developments in psychometrics (pp. 471–478). Tokyo: Springer.
[27] van de Velden, M., & Kiers, H.A.L. (2005). Rotation in correspondence analysis. Journal of Classification, 22, 251–271. · Zbl 1336.62164
[28] Weisstein, E.W. (1998). CRC concise encyclopedia of mathematics. New York: CRC Press. · Zbl 1006.00006
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