K. van Deun
http://repub.eur.nl/ppl/1247/
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RePub, Erasmus University RepositoryVIPSCAL: 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.