XPCA：离散和连续变量组合的扩展PCA
XPCA: Extending PCA for a Combination of Discrete and Continuous Variables
摘要：
主成分分析（PCA）是多元探索性数据分析中最常用的分析工具。在本文中，我们考虑的问题是如何处理异构变量，包括连续，二元和序数。在低秩PCA的概率解释中，数据具有正态多变量分布，因此每列的边缘分布也是正态的。如果一些边缘是连续的，但不是正常的，半参数copula为基础的主成分分析（COCA）方法是一种替代PCA，结合高斯copula与非参数边缘。如果一些边缘是离散或半连续的，我们提出了一个新的扩展PCA（XPCA）方法，也使用高斯copula和非参数边缘和帐户的离散变量的似然计算通过在适当的间隔集成。与PCA一样，XPCA产生的因子可用于发现数据中的潜在结构，构建预测模型并进行降维。我们提出了新的模型，它的诱导似然函数，和拟合算法，可以应用在存在缺失数据。我们演示了如何使用XPCA为每个数据点生成估计的完整条件分布，并使用此来生成自动范围相关的缺失数据的估计值。我们比较的方法，适用于模拟和现实世界的数据集，具有离散和连续变量的混合物。
英文摘要：
Principal component analysis (PCA) is arguably the most popular tool in multivariate exploratory data analysis. In this paper, we consider the question of how to handle heterogeneous variables that include continuous, binary, and ordinal. In the probabilistic interpretation of low-rank PCA, the data has a normal multivariate distribution and, therefore, normal marginal distributions for each column. If some marginals are continuous but not normal, the semiparametric copula-based principal component analysis (COCA) method is an alternative to PCA that combines a Gaussian copula with nonparametric marginals. If some marginals are discrete or semi-continuous, we propose a new extended PCA (XPCA) method that also uses a Gaussian copula and nonparametric marginals and accounts for discrete variables in the likelihood calculation by integrating over appropriate intervals. Like PCA, the factors produced by XPCA can be used to find latent structure in data, build predictive models, and perform dimensionality reduction. We present the new model, its induced likelihood function, and a fitting algorithm which can be applied in the presence of missing data. We demonstrate how to use XPCA to produce an estimated full conditional distribution for each data point, and use this to produce to provide estimates for missing data that are automatically range respecting. We compare the methods as applied to simulated and real-world data sets that have a mixture of discrete and continuous variables.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
