具有方差极大的Vintage因子分析执行统计推断
Vintage Factor Analysis with Varimax Performs Statistical Inference
摘要：
心理学家开发了多因素分析，将多变量数据分解为少量可解释的因素，而无需任何关于这些因素的先验知识。在这种形式的因子分析中，方差最大“因子旋转”是使因子可解释的关键步骤。查尔斯·斯皮尔曼和许多其他人反对因子旋转，因为因子似乎是旋转不变的。这些反对意见仍然报告在所有当代多元统计教科书。这是一个工程，因为这种老式的因素分析形式已经幸存下来，并广泛流行，因为从经验上讲，因素轮换往往使因素更容易解释。我们认为，旋转使因素更容易解释，因为事实上，方差最大因子旋转执行统计推断。我们表明，主成分分析（PCA）与方差最大旋转提供了一个统一的谱估计策略，为广泛的一类现代因素模型，包括随机块模型和自然变化的潜在狄利克雷分配（即，“主题建模”）。此外，我们表明，瑟斯顿的广泛采用的稀疏诊断隐含评估一个关键的“尖峰”的条件，使旋转统计识别在这些模型。总的来说，这表明老式因素分析的诀窍执行统计推断，扭转了近一个世纪的统计思想的主题。使用稀疏特征值求解器，具有方差最大的PCA是快速和稳定的。结合Thurstone的直接诊断，这种老式的方法适用于各种现代应用。
英文摘要：
Psychologists developed Multiple Factor Analysis to decompose multivariate data into a small number of interpretable factors without any a priori knowledge about those factors. In this form of factor analysis, the Varimax "factor rotation" is a key step to make the factors interpretable. Charles Spearman and many others objected to factor rotations because the factors seem to be rotationally invariant. These objections are still reported in all contemporary multivariate statistics textbooks. This is an engima because this vintage form of factor analysis has survived and is widely popular because, empirically, the factor rotation often makes the factors easier to interpret. We argue that the rotation makes the factors easier to interpret because, in fact, the Varimax factor rotation performs statistical inference. We show that Principal Components Analysis (PCA) with the Varimax rotation provides a unified spectral estimation strategy for a broad class of modern factor models, including the Stochastic Blockmodel and a natural variation of Latent Dirichlet Allocation (i.e., "topic modeling"). In addition, we show that Thurstone's widely employed sparsity diagnostics implicitly assess a key "leptokurtic" condition that makes the rotation statistically identifiable in these models. Taken together, this shows that the know-how of Vintage Factor Analysis performs statistical inference, reversing nearly a century of statistical thinking on the topic. With a sparse eigensolver, PCA with Varimax is both fast and stable. Combined with Thurstone's straightforward diagnostics, this vintage approach is suitable for a wide array of modern applications.
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