Hoff Peter D
University of Washington.
Ann Appl Stat. 2015 Sep;9(3):1169-1193. doi: 10.1214/15-AOAS839. Epub 2015 Nov 2.
A fundamental aspect of relational data, such as from a social network, is the possibility of dependence among the relations. In particular, the relations between members of one pair of nodes may have an effect on the relations between members of another pair. This article develops a type of regression model to estimate such effects in the context of longitudinal and multivariate relational data, or other data that can be represented in the form of a tensor. The model is based on a general multilinear tensor regression model, a special case of which is a tensor autoregression model in which the tensor of relations at one time point are parsimoniously regressed on relations from previous time points. This is done via a separable, or Kronecker-structured, regression parameter along with a separable covariance model. In the context of an analysis of longitudinal multivariate relational data, it is shown how the multilinear tensor regression model can represent patterns that often appear in relational and network data, such as reciprocity and transitivity.
关系数据（如来自社交网络的数据）的一个基本方面是关系之间存在依赖的可能性。特别是，一对节点成员之间的关系可能会对另一对节点成员之间的关系产生影响。本文开发了一种回归模型，用于在纵向和多变量关系数据或其他可以用张量形式表示的数据背景下估计此类影响。该模型基于一般的多线性张量回归模型，其一个特殊情况是张量自回归模型，其中一个时间点的关系张量由前一时间点的关系简约回归得到。这是通过一个可分离的（即克罗内克结构的）回归参数以及一个可分离的协方差模型来实现的。在对纵向多变量关系数据的分析背景下，展示了多线性张量回归模型如何能够表示关系和网络数据中经常出现的模式，如互惠性和传递性。
