Abstract
In the present paper the ORTHOMAX rotation problem is reconsidered. It is shown that its solution can be presented as a steepest ascent flow on the manifold of orthogonal matrices. A matrix formulation of the ORTHOMAX problem is given as an initial value problem for matrix differential equation of first order. The solution can be found by any available ODE numerical integrator. Thus the paper proposes a convergent method for direct matrix solution of the ORTHOMAX problem.
The well-known first order necessary condition for the VARIMAX maximizer is reestablished for the ORTHOMAX case without using Lagrange multipliers. Additionally new second order optimality conditions are derived and as a consequence an explicit second order necessary condition for further classification of the ORTHOMAX maximizer is obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chu, M.T. & Driessel, K.R. (1990) The projected gradient method for least squares matrix approximations with spectral constraints, SI AM J. Numer. Anal., 27, 1050–1060.
Chu, M.T. (1991) A continuous Jacobi-like approach to the simultanious reduction of real matrices. Linear Algebra and Its Applications, 147, 75–96.
Chu, M.T. (1994) A list of matrix flows with applications, Fields Institute Communications, 3, 87–97.
Chu, M.T. & Trendafilov, N.T. (1996) On a differential equation approach to the weighted orthogonal procrustes problem, to appear in “Statistics & Computing”.
Gill, P.E., Murray, W., & Wright, M.K. (1981) Practical Optimization. Florida: Academic Press.
Helmke, U. & Moore, J.B. (1994) Optimization and Dynamical Systems, London: Springer Verlag.
Helmke, U. (1995) Isospectral matrix flows for numerical analysis, in Moonen, M. & De Moor, B. (Eds.) SVD and Signal Processing, III-Algorithms, Architectures and Applications, Amsterdam: Elsevier Science B.V.
Kaiser, H.F. (1958) The varimax criterion for analytic rotation in factor analysis, Psyckometrika, 23, 187–200.
Kiers, FLAX. (1990) Majorization as a tool for optimizing a class of matrix functions, Psychometrika, 55, 417–428.
Neudecker, H. (1981) On the matrix formulation of Kaiser’s Varimax criterion, Psychometrika, 46, 343–345.
Magnus, J.R. & Neudecker, H. (1988) Matrix Differential Calculus with Application in Statistics and Econometrics, New York: Wiley.
Mulaik, S.A. (1972) The Foundations of Factor Analysis, New York: McGraw-Hill.
Shampine, L.F. & Reichelt, M.W. (1995) The MATLAB ODE suite, preprint, (also available from anonymous ftp, mathworks, com:/pub/mathworks/toolbox/matlab/funfun)
ten Berge, J.M.F. (1984) A joint treatment of VARIMAX rotation and the problem of diagonalizing symmetric matrices simultaniously in the least-squares sense, Psychometrika, 49, 347–358.
ten Berge, J.M.F., Knol, D.L., & Kiers, H.A.L. (1988) A treatment of the ORTHOMAX rotation family in terms of diagonalization, and a re-examination of a singular value approach to VARIMAX rotation, Computational Statistics Quarterly, 3, 207–217.
Author information
Authors and Affiliations
Additional information
Proofs should be sent to Nickolay Trendafilov, Katholieke Universiteit Leuven, Department of Electrical Engineering (ESAT), Signals, Identification, System Theory and Automation (SISTA) Kardinaal Mercierlaan 94, B-3001 Leuven, BELGIUM.
This research was supported in part by National Science Foundation under grant DMS-9422280.
This research was performed while visiting the Department of Educational Psychology, Nagoya University, JAPAN and was supported by Japan Society for the Promotion of Science.
About this article
Cite this article
Chu, M.T., Trendafilov, N.T. Orthomax Rotation Problem. A Differential Equation Approach. Behaviormetrika 25, 13–23 (1998). https://doi.org/10.2333/bhmk.25.13
Received:
Published:
Issue Date:
DOI: https://doi.org/10.2333/bhmk.25.13