区间数据的因子分析
Factor Analysis of Interval Data
摘要：
本文提出了一个符号数据的因子分析模型，重点是区间值变量的特殊情况。所提出的方法描述了测量的区间值变量之间的相关结构，在一些基本的，但不可观察的，不相关的区间值变量，称为\textit{common factors}。在每个观察间隔内考虑均匀分布和三角分布。我们得到相应的样本均值，方差和协方差假设一般三角分布。在我们的建议中，因子提取主成分或主轴因子，区间值变量相关矩阵上执行。为了估计公共因子的值，通常称为\textit{factor scores}，考虑了两种方法，它们受到实值数据方法的启发：Bartlett方法和Anderson-Rubin方法。在这两种情况下，估计值是通过求解最小化分位数函数之间的加权平方马洛斯距离的函数的优化问题来获得的。分位数函数和平方马洛斯距离的显式表达式推导出假设一般三角分布。该方法的适用性说明使用两组数据：温度和降水在美国的城市之间的1971年和2000年和措施的汽车特性的不同品牌和型号。此外，该方法的合成数据与预定义的相关结构进行评估。
英文摘要：
This paper presents a factor analysis model for symbolic data, focusing on the particular case of interval-valued variables. The proposed method describes the correlation structure among the measured interval-valued variables in terms of a few underlying, but unobservable, uncorrelated interval-valued variables, called \textit{common factors}. Uniform and Triangular distributions are considered within each observed interval. We obtain the corresponding sample mean, variance and covariance assuming a general Triangular distribution. In our proposal, factors are extracted either by Principal Component or by Principal Axis Factoring, performed on the interval-valued variables correlation matrix. To estimate the values of the common factors, usually called \textit{factor scores}, two approaches are considered, which are inspired in methods for real-valued data: the Bartlett and the Anderson-Rubin methods. In both cases, the estimated values are obtained solving an optimization problem that minimizes a function of the weighted squared Mallows distance between quantile functions. Explicit expressions for the quantile function and the squared Mallows distance are derived assuming a general Triangular distribution. The applicability of the method is illustrated using two sets of data: temperature and precipitation in cities of the United States of America between the years 1971 and 2000 and measures of car characteristics of different makes and models. Moreover, the method is evaluated on synthetic data with predefined correlation structures.
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