Understanding and controlling rotations in factor analytic models
Understanding and controlling rotations in factor analytic models
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DOI：
10.1016/s0169-7439(01)00200-3
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发表时间：
2002-01-28
影响因子：
3.9
通讯作者：
Ramadan, Z
中科院分区：
文献类型：
--
作者：
Paatero, P;Hopke, PK;Ramadan, Z
正矩阵分解（PMF）是求解因子分析问题的最小二乘方法。它已以几种形式实现。最初，使用了一个名为PMF2的程序。随后，一个新的，更灵活的建模工具，多线性引擎，被开发。这些程序可以利用不同的方法来处理旋转不确定性问题。虽然两者都利用非负性约束来降低转动自由度，但这些约束通常不足以完全消除转动问题。本文讨论了控制旋转的其他方法：(1)在“分数”之间整体施加加法和相应“负载”之间的减法（反之亦然），(2)约束单个因子元素，分数和/或负载，趋向于零值，(3)规定某些关键因子元素的比率值，或(4)指定加载矩阵的某些列为已知的固定值。需要强调的是，这些技术的应用必须以一些关于可接受的或理想的因素形状的外部信息为基础。如果不存在这样的先验信息，那么可以探索所有可能的旋转，但是，没有根据选择这些旋转中的一个作为“最佳”结果。讨论了在任何特定结果中估计旋转模糊度的方法。(C) 2002 Elsevier Science B.V.版权所有
Positive Matrix Factorization (PMF) is a least-squares approach for solving the factor analysis problem. It has been implemented in several forms. Initially, a program called PMF2 was used. Subsequently, a new, more flexible modeling tool, the Multilinear Engine, was developed. These programs can utilize different approaches to handle the problem of rotational indeterminacy. Although both utilize non-negativity constraints to reduce rotational freedom, such constraints are generally insufficient to wholly eliminate the rotational problem. Additional approaches to control rotations are discussed in this paper: (1) global imposition of additions among "scores" and subtractions among the corresponding "loadings" (or vice versa), (2) constraining individual factor elements, either scores and/or loadings, toward zero values, (3) prescribing values for ratios of certain key factor elements, or (4) specifying certain columns of the loadings matrix as known fixed values. It is emphasized that application of these techniques must be based on some external information about acceptable or desirable shapes of factors. If no such a priori information exists, then the full range of possible, rotations can be explored, but, there is no basis for choosing one of these rotations as the "best" result. Methods for estimating the rotational ambiguity in any specific result are discussed. (C) 2002 Elsevier Science B.V. All rights reserved.