Simplicity of core arrays in three-way principal component analysis and the typical rank of p x q x 2 arrays
Simplicity of core arrays in three-way principal component analysis and the typical rank of p x q x 2 arrays
复制标题
DOI：
10.1016/s0024-3795(99)00057-9
复制
发表时间：
1999-06-15
影响因子：
1.1
通讯作者：
Kiers, HAL
中科院分区：
文献类型：
--
作者：
Ten Berge, JMF;Kiers, HAL
当核心阵列具有大量零元素时，三向阵列的主成分分析解的解释将大大简化。实现这一目标的可能性最近一方面通过转向简单或简单的目标，另一方面通过数学分析进行了探索。本文证明了一个pxqx2数组，当pb> q大于等于2时，几乎可以肯定地变换为除2q个元素外所有元素为零。还证明了这种形式的数组的三阶秩最多为p。这对pxqx2数组的典型秩有直接影响，当p = q时也是如此。当p大于或等于2q时，典型秩为2q；当q < p < 2q时，它是p，当p = q时，秩通常（几乎肯定）是p或p + 1。这些典型的秩结果适用于用实值秩1数组分解实值三向数组，而不适用于复杂设置，其中p x q x 2数组的典型秩在p > q时也是min[p，2q]，但在p = q时是p。(C) 1999 Elsevier Science Inc.。版权所有。
Interpreting the solution of a Principal Component Analysis of a three-way array is greatly simplified when the core array has a large number of zero elements. The possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand, and by mathematical analysis on the other. In the present paper, it is shown that a p x q x 2 array, with p > q greater than or equal to 2, can almost surely be transformed to have all but 2q elements zero. It is also shown that arrays of that form have three-way rank p at most. This has direct implications for the typical rank of p x q x 2 arrays, also when p = q. When p greater than or equal to 2q, the typical rank is 2q; when q < p < 2q it is p, and when p = q, the rank is typically (almost surely) p or p + 1. These typical rank results pertain to the decomposition of real valued three-way arrays in terms of real valued rank one arrays, and do not apply in the complex setting, where the typical rank of p x q x 2 arrays is also min[p,2q] when p > q, but it is p when p = q. (C) 1999 Elsevier Science Inc. All rights reserved.