A. J. Kearsley, R. A. Tapia, and M. W. Trosset. The solution of the metric STRESS and SSTRESS problems in multidimensional scaling using Newton's method. Computational Statistics, 13:369-396, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....places where points coincide. The majorization method can be seen as a gradient method with a constant step size, which uses convex analysis methods to prove convergence. More recently, faster linearly or superlinearly convergent methods have been tried successfully (Glunt, Hayden Rayden 1993, Kearsley, Tapia Trosset 1998). One of the key advantages of the majorization method is that it extends easily to restricted MDS problems (de Leeuw Heiser 1980b) Each subproblem in the sequence is a least squares projection problem on the set of configurations satisfying the constraints, which is usually easy to solve. ....

....through the coordinates were alternated with isotonic regressions in the nonmetric case. More efficient alternating least squares algorithms were developed later by De Leeuw, Takane, and Browne (cf. Browne (1987) and superlinear and quadratic methods were proposed by W. Glunt Liu (1991) and Kearsley et al. 1998). 5.3. STRAIN. Minimizing STRAIN was, and is, the preferred algorithm in metric MDS. It is also used as the starting point in iterative nonmetric algorithms. Recently, more general algorithms for minimizing STRAIN in nonmetric and distance completion scaling have been proposed by Trosset (1998b) ....

Kearsley, A., Tapia, R. & Trosset, M. (1998), `The solution of the metric STRESS and SSTRESS problems in multidimensional scaling using Newton's method', Computational Statistics 13, 369--396.

.... r , metric MDS attempts to nd a con guration matrix X that minimizes r . Unfortunately, minimizers must be computed by an iterative algorithm for numerical optimization. A survey of some of the more ecient algorithms for minimizing 1 and 2 was made by Kearsley, Tapia, and Trosset [12]. These algorithms nd local minimizers. To improve the chance that the algorithm will converge to a global minimizer, the user is advised to choose a good initial con guration. The conventional choice the default initial con guration used in many implementations is the con guration constructed ....

....stress or sstress can be trapped by nonglobal minimizers. Some researchers, e.g. Groenen [7] and Groenen and Heiser [8] argued that nonglobal minimizers of the stress criterion are common and recommended global optimization algorithms for metric MDS. In contrast, Kearsley, Tapia, and Trosset [12] demonstrated that the popular SMACOF algorithm [10] for minimizing stress tends to terminate prematurely, creating the false impression that a local minimizer has been found. They recommended more sophisticated, second order methods for metric MDS. Because the only evidence for nonglobal ....

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A. J. Kearsley, R. A. Tapia, and M. W. Trosset. The solution of the metric STRESS and SSTRESS problems in multidimensional scaling using Newton's method. Computational Statistics, 13:369-396, 1998.

....selected an error criterion oe r , metric MDS attempts to find a configuration matrix X that minimizes oe r . Unfortunately, minimizers must be computed by an iterative algorithm for numerical optimization. A survey of some of the more efficient algorithms for minimizing oe 1 and oe 2 was made by Kearsley, Tapia, and Trosset (1998). These algorithms find local minimizers. To improve the chance that the algorithm will converge to a global minimizer, the user is advised to choose a good initial configuration. The conventional choice the default initial configuration used in many implementations is the configuration ....

Kearsley, A. J., Tapia, R. A., and Trosset, M. W. (1998). The solution of the metric STRESS and SSTRESS problems in multidimensional scaling using Newton's method. Computational Statistics, 13(3):369--396.

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A. J. Kearsley, R. A. Tapia, and M. Trosset, The solution of the metric stress and sstress problems in multidimensional scaling using Newtons method, Computational Statistics, 13 (1998), pp. 369--396.

....variables for the purpose of precluding degenerate solutions. Finally, we impose structural constraints on the configuration matrix that remove translational and rotational indeterminancies. The importance of imposing such constraints, as well as our technique for imposing them, was discussed by Kearsley, Tapia, and Trosset (1998). All of the above considerations can be easily specified using AMPL (A Mathematical Programming Language) documented by Fourer, Gay, and Kernighan (1993) Moreover, AMPL will detect separable structure and calculate derivatives by automatic differentiation before passing the problem to a ....

Kearsley, A. J., Tapia, R. A., and Trosset, M. W. (1998). The solution of the metric STRESS and SSTRESS problems in multidimensional scaling using Newton's method. Computational Statistics, 13(3):369--396.

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