论文标题
高阶因子分析模型的维度
Dimensions of Higher Order Factor Analysis Models
论文作者
论文摘要
因子分析模型是一个统计模型，其中一定数量的隐藏随机变量（称为因子）线性地影响另一组观察到的随机变量的行为，并具有附加的随机噪声。该模型的主要假设是因子和噪声是高斯随机变量。这意味着可行的集合在于阳性半芬属矩阵的锥体中。在本文中，我们不认为因子和噪声是高斯，因此，观察到的变量的高阶力矩和累积张量通常为非零。这激发了KTH阶因子分析模型的概念，即在一个因子分析模型中所有随机向量的家族，其中因子和噪声具有有限的，可能是非零的力矩，并且累积张量张紧到订单k。该子集可以被描述为多项式图的图像上对称张量空间的笛卡尔产物。我们的目标是计算其尺寸，我们提供图像具有正编码的条件。
The factor analysis model is a statistical model where a certain number of hidden random variables, called factors, affect linearly the behaviour of another set of observed random variables, with additional random noise. The main assumption of the model is that the factors and the noise are Gaussian random variables. This implies that the feasible set lies in the cone of positive semidefinite matrices. In this paper, we do not assume that the factors and the noise are Gaussian, hence the higher order moment and cumulant tensors of the observed variables are generally nonzero. This motivates the notion of kth-order factor analysis model, that is the family of all random vectors in a factor analysis model where the factors and the noise have finite and possibly nonzero moment and cumulant tensors up to order k. This subset may be described as the image of a polynomial map onto a Cartesian product of symmetric tensor spaces. Our goal is to compute its dimension and we provide conditions under which the image has positive codimension.
