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Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation

机译：通过惩罚变异系数避免多维展开中的简并性

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摘要

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.
机译：多维展开方法几乎在所有情况下都存在退化问题。大多数简并性很容易识别：这些解决方案是完美的，但琐碎，其特点是来自不同集合的点之间的距离大致相等。提出了绝对退化解的定义，该定义明确指出，仅当变换函数中存在截距时，才会出现这些解。已经提出并测试了许多关于退化问题的解决方案，但迄今为止收效甚微。在本文中，我们对一种方法进行了实质性的修改，该方法是基于通常的最小二乘损失函数的方差引入了归一化因子。 Heiser未发表论文，（1981年），并表明Kruskal和Carroll提出的归一化因子不足以避免退化。本文提出的因子基于变异系数，阻止或惩罚具有小变异的邻近区域的非度量变换，从而使该过程避开了点间距离具有小变异的解。描述了一种基于迭代主化最小化重新调整损失函数的算法。讨论了仿真研究的结果，其中确定了惩罚参数的最佳范围。我们的方法分析了两个经验数据集，清楚地表明了所提出的损失函数的好处。

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