XPCA

Xpca, short for eXtended Principal Component Analysis, is a family of dimensionality-reduction methods that generalize classical principal component analysis (PCA) to data that do not follow a Gaussian distribution. XPCA adopts a probabilistic latent-variable framework in which observed entries arise from a distribution in the exponential family conditioned on low-dimensional latent factors. The data matrix is modeled as the sum of a low-rank signal and noise, with the latent factors capturing the major structure.

Estimation typically proceeds via maximum likelihood or Bayesian inference, using algorithms such as expectation-maximization or alternating optimization over the latent factors and the loading matrix. The approach handles missing values naturally and can incorporate different observation models by selecting an appropriate distribution and link function for the data type (binary, count, ordinal, or continuous).

XPCA is particularly suited to datasets where Gaussian assumptions are violated, such as survey responses with ordinal scales, RNA-sequencing counts, or user-item interaction data. By aligning the observation model with the data type, XPCA can yield more accurate reconstructions and interpretable components compared with standard PCA or singular-value decomposition.

Choosing the number of components is typically done by cross-validation or information criteria; the method is more computationally intensive than PCA and requires careful specification of the distribution family. It is often used as a generalization of PCA within generalized linear and probabilistic modeling frameworks.

XPCA relates to broader generalized PCA approaches and probabilistic PCA variants, and it complements matrix factorization methods used in collaborative filtering and data imputation.