Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, QC, H3A 1B1 , Canada.
University of Groningen, Groningen, The Netherlands.
Psychometrika. 2018 Mar;83(1):1-20. doi: 10.1007/s11336-017-9558-9. Epub 2017 Feb 14.
Parallel factor analysis (PARAFAC) is a useful multivariate method for decomposing three-way data that consist of three different types of entities simultaneously. This method estimates trilinear components, each of which is a low-dimensional representation of a set of entities, often called a mode, to explain the maximum variance of the data. Functional PARAFAC permits the entities in different modes to be smooth functions or curves, varying over a continuum, rather than a collection of unconnected responses. The existing functional PARAFAC methods handle functions of a one-dimensional argument (e.g., time) only. In this paper, we propose a new extension of functional PARAFAC for handling three-way data whose responses are sequenced along both a two-dimensional domain (e.g., a plane with x- and y-axis coordinates) and a one-dimensional argument. Technically, the proposed method combines PARAFAC with basis function expansion approximations, using a set of piecewise quadratic finite element basis functions for estimating two-dimensional smooth functions and a set of one-dimensional basis functions for estimating one-dimensional smooth functions. In a simulation study, the proposed method appeared to outperform the conventional PARAFAC. We apply the method to EEG data to demonstrate its empirical usefulness.
平行因子分析（PARAFAC）是一种有用的多元方法，可用于同时分解由三种不同类型实体组成的三向数据。该方法估计三次线性分量，每个分量都是一组实体的低维表示，通常称为模态，以解释数据的最大方差。功能 PARAFAC 允许不同模式中的实体成为平滑函数或曲线，在连续体上变化，而不是一组不相关的响应。现有的功能 PARAFAC 方法仅处理一维参数（例如时间）的函数。在本文中，我们提出了一种新的功能 PARAFAC 扩展，用于处理其响应沿二维域（例如具有 x 和 y 轴坐标的平面）和一维参数排序的三向数据。从技术上讲，所提出的方法将 PARAFAC 与基函数扩展逼近相结合，使用一组分段二次有限元基函数来估计二维平滑函数和一组一维基函数来估计一维平滑函数。在模拟研究中，所提出的方法似乎优于传统的 PARAFAC。我们将该方法应用于 EEG 数据，以证明其实际有用性。
