  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.
