K. van Deun
http://repub.eur.nl/ppl/1247/
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http://repub.eur.nl/
RePub, Erasmus University RepositoryMajorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi
http://repub.eur.nl/pub/74748/
Tue, 01 Nov 2005 00:00:01 GMT<div>K. van Deun</div><div>P.J.F. Groenen</div>
Interpreting degenerate solutions in unfolding by use of the vector model and the compensatory distance model
http://repub.eur.nl/pub/64034/
Tue, 01 Mar 2005 00:00:01 GMT<div>K. van Deun</div><div>P.J.F. Groenen</div><div>W.J. Heiser</div><div>F.M.T.A. Busing</div><div>L. Delbeke</div>
In this paper, we reconsider the merits of unfolding solutions based on loss functions involving a normalization on the variance per subject. In the literature, solutions based on Stress-2 are often diagnosed to be degenerate in the majority of cases. Here, the focus lies on two frequently occurring types of degeneracies. The first type typically locates some subject points far away from a compact cluster of the other points. In the second type of solution, the object points lie on a circle. In this paper, we argue that these degenerate solutions are well fitting and informative. To reveal the information, we introduce mixtures of plots based on the ideal point model of unfolding, the vector model, and on the signed distance model. In addition to a different representation, we provide a new iterative majorization algorithm to optimize the average squared correlation between the distances in the configuration and the transformed data per individual. It is shown that this approach is equivalent to minimizing Kruskal's Stress-2.VIPSCAL: A combined vector ideal point model for preference data
http://repub.eur.nl/pub/1904/
Thu, 20 Jan 2005 00:00:01 GMT<div>K. van Deun</div><div>P.J.F. Groenen</div><div>L. Delbeke</div>
In this paper, we propose a new model that combines the vector model and the
ideal point model of unfolding. An algorithm is developed, called VIPSCAL, that
minimizes the combined loss both for ordinal and interval transformations. As such,
mixed representations including both vectors and ideal points can be obtained but
the algorithm also allows for the unmixed cases, giving either a complete ideal
pointanalysis or a complete vector analysis. On the basis of previous research,
the mixed representations were expected to be nondegenerate. However, degenerate
solutions still occurred as the common belief that distant ideal points can be represented by vectors does not hold true. The occurrence of these distant ideal points was solved by adding certain length and orthogonality restrictions on the configuration. The restrictions can be used both for the mixed and unmixed cases in several ways such that a number of different models can be fitted by VIPSCAL.Majorization algorithms for inspecting circles, ellipses, squares, rectangles, and rhombi
http://repub.eur.nl/pub/944/
Fri, 26 Sep 2003 00:00:01 GMT<div>K. van Deun</div><div>P.J.F. Groenen</div>
In several disciplines, as diverse as shape analysis, location
theory, quality control, archaeology, and psychometrics, it can be
of interest to fit a circle through a set of points. We use the
result that it suffices to locate a center for which the variance
of the distances from the center to a set of given points is
minimal. In this paper, we propose a new algorithm based on
iterative majorization to locate the center. This algorithm is
guaranteed to yield a series nonincreasing variances until a
stationary point is obtained. In all practical cases, the
stationary point turns out to be a local minimum. Numerical
experiments show that the majorizing algorithm is stable and fast.
In addition, we extend the method to fit other shapes, such as a
square, an ellipse, a rectangle, and a rhombus by making use of
the class of $l_p$ distances and dimension weighting. In addition,
we allow for rotations for shapes that might be rotated in the
plane. We illustrate how this extended algorithm can be used as a
tool for shape recognition.