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Newton Algorithms for Analytic Rotation: an Implicit Function Approach

Published online by Cambridge University Press:  01 January 2025

Robert J. Boik
Robert J. Boik*
Affiliation:
Montana State University
*
Request for reprints should be sent to Robert J. Boik, Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717-2400, USA. E-mail: [email protected]
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Abstract

In this paper implicit function-based parameterizations for orthogonal and oblique rotation matrices are proposed. The parameterizations are used to construct Newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables. The speed of the new algorithms is compared to that of existing algorithms and to that of Newton algorithms based on alternative parameterizations. Several rotation criteria were examined and the algorithms were evaluated over a range of values for m. Initial guesses for Newton algorithms were improved by subconvergence iterations of the gradient projection algorithm. Simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m. Among competing algorithms, the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large. In all other conditions, however, the implicit function algorithms were competitive with or faster than the fastest existing algorithms. The new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion.

Keywords

component loss criterionfactor analysisgradient projection algorithmoblique rotationorthogonal rotationorthogonal matrixplanar algorithmprincipal components
Type
Theory and Methods
Information
Psychometrika , Volume 73 , Issue 2 , June 2008 , pp. 231 - 259
Copyright
Copyright © 2007 The Psychometric Society

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