Principal Component Pursuit for Pattern Identification in Environmental Mixtures

Abstract

Background: Environmental health researchers often aim to identify sources or behaviors that give rise to potentially harmful environmental exposures. Objective: We adapted principal component pursuit (PCP)—a robust and well-established 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 patterns of exposure across pollutants and a sparse matrix isolating unique or extreme exposure events. Methods: We adapted PCP to accommodate nonnegative data, missing data, and values below a given limit of detection (LOD). We simulated data to represent environmental mixtures of two sizes with increasing proportions and three noise structures. We applied PCP-LOD to evaluate its performance in comparison with principal component analysis (PCA). We next applied principal component pursuit with limit of detection (PCP-LOD) to an exposure mixture of 21 persistent organic pollutants (POPs) measured in 1,000 U.S. adults from the 2001–2002 National Health and Nutrition Examination Survey (NHANES). We applied singular value decomposition to the estimated low-rank matrix to characterize the patterns. Results: 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 data sets with up to 50% of the data . When 75% of values were , 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 extreme events. One pattern represented comprehensive exposure to all POPs. The other patterns grouped chemicals based on known structure and toxicity. Discussion: PCP-LOD serves as a useful tool to express multidimensional exposures as consistent patterns that, if found to be related to adverse health, are amenable to targeted public health messaging. https://doi.org/10.1289/EHP10479

Applying principal component pursuit to investigate the association between source-specific fine particulate matter and myocardial infarction hospitalizations in New York City

“…Finally, we found that when applying nonconvex falsePCP to speciated PM 2.5 data, it was necessary to tune hyperparameters, which is a time-intensive process that comes with a certain degree of researcher subjectivity. Other formulations of PCP have used theoretically-optimal single universal values for hyperparameters falseλ and falseμ, 17,18 but we found that these approaches were not flexible enough to detect the underlying patterns present in our dataset, as they require a better-defined low-rank structure.…”

“…We used square-root PCP (falsePCP), an extension of PCP, 18 and combined it with a separate extension introducing a nonconvex penalty on the low-rank matrix. 17 We used cross-validation to select the optimal rank of the low-rank matrix, which can be understood as the number of underlying patterns in the PM 2.5 data. Please see the Supplement for further details on hyperparameter selection.…”

Invited Perspective: Challenges and Opportunities for Missing Data in the Context of Environmental Mixture Methods

“…In simulations, PCP-LOD generally outperformed PCA (e.g., PCP-LOD recovered a higher percentage of the true number of patterns) when the percentage of observations below the LOD was and in scenarios in which there was either low Gaussian noise or there were both low Gaussian noise and sparse events. 10 Further, PCP-LOD largely outperformed PCA when 16 chemicals were included in the mixture, but performance decreased when the number of chemicals in the mixture increased to 48. In the application to NHANES data, PCP-LOD produced results similar to those of PCA when applied to a mixture of 21 chemicals (including dioxins, furans, and polychlorinated biphenyls) with above the LOD.…”

“…In their new study, 10 Gibson et al. adapted principal component pursuit (PCP), a robust method for dimensionality reduction and pattern identification, to accommodate missing data and values below the LOD.…”

“…In the application to NHANES data, PCP-LOD produced results similar to those of PCA when applied to a mixture of 21 chemicals (including dioxins, furans, and polychlorinated biphenyls) with above the LOD. 10 …”