Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis
Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis
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发表时间：
1970
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通讯作者：
R. Harshman
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作者：
R. Harshman
简单的结构和其他常见的因素轮换原则一般不提供强有力的理由，归因于他们选择的因素的解释意义。与此相反，它示出的Cattell的旋转成比例的配置文件（PP）的原则的扩展提供了一个基础，用于确定三路或更高阶的多模式数据的解释因素。针对多模式数据变异的两种基本模式--系统变异和对象变异建立了概念模型，并发现PP分析适用于系统变异的情况。虽然PP最初被制定为旋转的原则与经典的双向因子分析，它被证明是体现一个潜在的三模式因子模型，这是在这里明确和广义的皱眉两到N“平行场合”。如最初制定的，PP旋转被限制为正交因子。广义PP模型被证明是唯一的“正确”的解决方案，斜，非简单的结构，甚至非线性因子结构。一系列的测试，与已知的因素组成的合成数据进行，展示了线性和非线性版本的模型的能力，提供数据的最小必要条件的唯一性，并揭示了这些最小条件不满足时的分析程序的属性。此外，给出了在一定条件下解的唯一性的数学证明。三模式PP因子分析是应用到一个三路的真实的数据，包括基本的和前三个共振峰频率的11人说8个元音。一个独特的解决方案被提取出来，由三个因素，这是非常有意义的，并与有关元音质量的先验知识和理论相一致。探讨了三模态PP模型与Tucker的多模态模型、McDonald的非线性模型以及卡罗尔和Chang的多维尺度模型之间的关系。
Simple structure and other common principles of factor rotation do not in general provide strong grounds for attributing explanatory significance to the factors which they select. In contrast, it is shown that an extension of Cattell's principle of rotation to Proportional Profiles (PP) offers a basis for determining explanatory factors for three-way or higher order multi-mode data. Conceptual models are developed for two basic patterns of multi-mode data variation, systemand object-variation, and PP analysis is found to apply in the system-variation case. Although PP was originally formulated as a principle of rotation to be used with classic two-way factor analysis, it is shown to embody a latent three-mode factor model, which is here made explicit and generalized frown two to N "parallel occasions". As originally formulated, PP rotation was restricted to orthogonal factors. The generalized PP model is demonstrated to give unique "correct" solutions with oblique, non-simple structure, and even non-linear factor structures. A series of tests, conducted with synthetic data of known factor composition, demonstrate the capabilities of linear and non-linear versions of the model, provide data on the minimal necessary conditions of uniqueness, and reveal the properties of the analysis procedures when these minimal conditions are not fulfilled. In addition, a mathematical proof is presented for the uniqueness of the solution given certain conditions on the data. Three-mode PP factor analysis is applied to a three-way set of real data consisting of the fundamental and first three formant frequencies of 11 persons saying 8 vowels. A unique solution is extracted, consisting of three factors which are highly meaningful and consistent with prior knowledge and theory concerning vowel quality. The relationships between the three-mode PP model and Tucker's multi-modal model, McDonald's non-linear model and Carroll and Chang's multi-dimensional scaling model are explored.