  Independent component analysis (ICA) is popular in many applications,
including cognitive neuroscience and signal processing. Due to computational
constraints, principal component analysis is used for dimension reduction prior
to ICA (PCA+ICA), which could remove important information. The problem is that
interesting independent components (ICs) could be mixed in several principal
components that are discarded and then these ICs cannot be recovered. We
formulate a linear non-Gaussian component model with Gaussian noise components.
To estimate this model, we propose likelihood component analysis (LCA), in
which dimension reduction and latent variable estimation are achieved
simultaneously. Our method orders components by their marginal likelihood
rather than ordering components by variance as in PCA. We present a parametric
LCA using the logistic density and a semi-parametric LCA using tilted Gaussians
with cubic B-splines. Our algorithm is scalable to datasets common in
applications (e.g., hundreds of thousands of observations across hundreds of
variables with dozens of latent components). In simulations, latent components
are recovered that are discarded by PCA+ICA methods. We apply our method to
multivariate data and demonstrate that LCA is a useful data visualization and
dimension reduction tool that reveals features not apparent from PCA or
PCA+ICA. We also apply our method to an fMRI experiment from the Human
Connectome Project and identify artifacts missed by PCA+ICA. We present
theoretical results on identifiability of the linear non-Gaussian component
model and consistency of LCA.
