  Environmental health researchers often aim to identify sources/behaviors that
give rise to potentially harmful exposures. We adapted principal component
pursuit (PCP)-a robust technique for dimensionality reduction in computer
vision and signal processing-to identify patterns in environmental mixtures.
PCP decomposes the exposure mixture into a low-rank matrix containing
consistent exposure patterns across pollutants and a sparse matrix isolating
unique exposure events. We adapted PCP to accommodate non-negative and missing
data, and values below a given limit of detection (LOD). We simulated data to
represent environmental mixtures of two sizes with increasing proportions <LOD
and three noise structures. We compared PCP-LOD to principal component analysis
(PCA) to evaluate performance. We next applied PCP-LOD to a mixture of 21
persistent organic pollutants (POPs) measured in 1,000 U.S. adults from the
2001-2002 National Health and Nutrition Examination Survey. We applied singular
value decomposition to the estimated low-rank matrix to characterize the
patterns. PCP-LOD recovered the true number of patterns through
cross-validation for all simulations; based on an a priori specified criterion,
PCA recovered the true number of patterns in 32% of simulations. PCP-LOD
achieved lower relative predictive error than PCA for all simulated datasets
with up to 50% of the data <LOD. When 75% of values were <LOD, PCP-LOD
outperformed PCA only when noise was low. In the POP mixture, PCP-LOD
identified a rank-three underlying structure and separated 6% of values as
unique events. One pattern represented comprehensive exposure to all POPs. The
other patterns grouped chemicals based on known structure and toxicity. PCP-LOD
serves as a useful tool to express multi-dimensional exposures as consistent
patterns that, if found to be related to adverse health, are amenable to
targeted interventions.
