Computational Statistics & Data Analysis 50 (2006) 135 – 153 www.elsevier.com/locate/csda Sensitivity analysis of the strain criterion for multidimensional scaling R.M. Lewis, M.W. Trosset ∗ Department of Mathematics, College ofWilliam & Mary, P.O. Box 8795,Williamsburg, VA 23185-8795, USA Available online 18 August 2004 Abstract Multidimensional scaling (MDS) is a collection of data analytic techniques for constructing con- figurations of points from dissimilarity information about interpoint distances. Classsical MDS as- sumes a fixed matrix of dissimilarities. However, in some applications, e.g., the problem of inferring 3-dimensional molecular structure from bounds on interatomic distances, the dissimilarities are free to vary, resulting in optimization problems with a spectral objective function. A perturbation analysis is used to compute first- and second-order directional derivatives of this function. The gradient and Hessian are then inferred as representers of the derivatives. This coordinate-free approach reveals the matrix structure of the objective and facilitates writing customized optimization software. Also analyzed is the spectrum of the Hessian of the objective. © 2004 Elsevier B.V.All rights reserved. Keywords: Classical multidimensional scaling; Principal coordinate analysis; Distance matrices; Distance geometry; Spectral decomposition; Perturbation analysis 1. Introduction Multidimensional scaling (MDS) is a collection of data analytic techniques for con- structing configurations of points from dissimilarity information about interpoint distances. Developed primarily by psychometricians and statisticians, MDS is widely used in a variety of disciplines for visualization and dimension reduction. The extensive literature on MDS ∗ Corresponding author. Tel.: +1-757-221-2040; fax: +1-757-221-7400. E-mail addresses: buckaroo@math.wm.edu (R.M. Lewis), trosset@math.wm.edu (M.W. Trosset) URLs: http://www.math.wm.edu/˜buckaroo, http://www.math.wm.edu/∼trosset. 0167-9473/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2004.07.011