Principal points of a multivariate mixture distribution
Principal points of a multivariate mixture distribution
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DOI：
10.1016/j.jmva.2010.08.009
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发表时间：
2011-02
期刊：
影响因子：
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通讯作者：
S. Matsuura;H. Kurata
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文献类型：
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作者：
S. Matsuura;H. Kurata
一个分布的n-主点集被定义为一个以均方距离最优表示该分布的n个点的集合。它提供了分布的最佳n点近似。然而，通常很难找到一组多元分布的主点。Tarpey等人[T.塔佩湖李，B。Flury，椭圆分布的主点和自洽点，Ann. Statist。23（1995）103-112]建立了一个定理，该定理指出椭圆对称分布的任何n-主点集都在由协方差矩阵的一些主特征向量所张成的线性子空间中。这个定理称为“主子空间定理”，是计算主点的有力工具。在实践中，我们经常遇到由几个子群组成的分布。因此，它是感兴趣的，知道是否主子空间定理仍然有效，即使在这种复杂的分布。在本文中，我们定义了一个多元位置混合模型。建立了一个定理，阐明了一个线性子空间中存在n-主点。
A set of n-principal points of a distribution is defined as a set of n points that optimally represent the distribution in terms of mean squared distance. It provides an optimal n-point-approximation of the distribution. However, it is in general difficult to find a set of principal points of a multivariate distribution. Tarpey et al. [T. Tarpey, L. Li, B. Flury, Principal points and self-consistent points of elliptical distributions, Ann. Statist. 23 (1995) 103–112] established a theorem which states that any set of n-principal points of an elliptically symmetric distribution is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem, called a “principal subspace theorem”, is a strong tool for the calculation of principal points. In practice, we often come across distributions consisting of several subgroups. Hence it is of interest to know whether the principal subspace theorem remains valid even under such complex distributions. In this paper, we define a multivariate location mixture model. A theorem is established that clarifies a linear subspace in which n-principal points exist.