In Columbia-PRIME/pcpr: Principal Component Pursuit for Environmental Epidemiology

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pcpr

Overview

The R package pcpr implements Principal Component Pursuit (PCP), a robust dimensionality reduction technique, for pattern recognition tailored to environmental health data. The statistical methodology and computational details are provided in Gibson et al. (2022).

Installation

You can install the latest official CRAN release of pcpr with:

install.packages("pcpr")

The development version of pcpr can be installed from GitHub with:

# install.packages("pak")
pak::pak("Columbia-PRIME/pcpr")

pcpr can then be loaded and attached in your current R session as usual with

library(pcpr)

Getting help

Extensive documentation is available on our pkgdown website and offline within R. You can see the pcpr reference manual in R with:

help("pcpr")

A number of vignettes are available from within R. They can be browsed using:

browseVignettes("pcpr")

We recommend reading the vignettes in the following order:

1. Theory crash course, or if directly in R: vignette("theory-crash-course")
2. Quickstart, or if directly in R: vignette("pcp-quickstart")
3. Air pollution source apportionment with PCP, or if directly in R: vignette("pcp-applied")

Have a bug to report or question to ask? Open an issue on our GitHub.

Modeling overview

PCP algorithms model an observed exposure matrix $D$ as the sum of three underlying ground-truth matrices:

a low-rank matrix $L_0$ encoding consistent patterns of exposure, a sparse matrix $S_0$ isolating unique or outlying exposure events (that cannot be explained by the consistent exposure patterns), and dense noise $Z_0$.

The models in pcpr seek to decompose an observed data matrix D into estimated low-rank and sparse components L and S for use in downstream environmental health analyses. The functions in pcpr are outfitted with three environmental health (EH)-specific extensions making pcpr particularly powerful for EH research:

1. Missing value functionality
2. Leveraging potential limit of detection (LOD) information
3. Non-negativity constraint on the estimated L matrix

PCP in environmental health studies

The methods in pcpr have already been applied in many environmental health studies. Several are listed below:

• Tao et al. (2023) apply PCP to investigate the association between source-specific fine particulate matter and myocardial infarction hospitalizations in NYC.
• Wu et al. (2024) employ PCP for exposome profiling of environmental pollutants in seminal plasma, uncovering novel associations with semen parameters.
• Benavides et al. (2024) use PCP to develop a Community Severity Index in NYC, measuring the barrier effect of road infrastructure and traffic in cities.

Acknowledgements

Please cite use of pcpr with:

Chillrud L, Benavides J, Gibson E, Zhang J, Yan J, Wright J, Goldsmith J, Kioumourtzoglou M (2025). pcpr: Principal Component Pursuit for Environmental Epidemiology. R package version 1.0.0, https://columbia-prime.github.io/pcpr/, https://github.com/Columbia-PRIME/pcpr.

@Manual{,
  title = {pcpr: Principal Component Pursuit for Environmental Epidemiology},
  author = {Lawrence G. Chillrud and Jaime Benavides and Elizabeth A. Gibson and Junhui Zhang and Jingkai Yan and John N. Wright and Jeff Goldsmith and Marianthi-Anna Kioumourtzoglou},
  year = {2025},
  note = {R package version 1.0.0, https://github.com/Columbia-PRIME/pcpr},
  url = {https://columbia-prime.github.io/pcpr/},
}

Please also cite Gibson et al. (2022).

This work was supported by NIEHS PRIME R01 ES028805.

Special thanks to Sophie Calhoun for designing pcpr's logo!

Usage

# In the below example, we simulate a simple mixtures model and run PCP,
# comparing it's performance to that of PCA. For an in depth example with
# simulated data, see vignette("pcp-quickstart"). For more realistic
# PCP usage, check out vignette("pcp-applied").

# Simulate an environmental mixture
data <- sim_data(
  n = 100, p = 10, r = 3,
  sparse_nonzero_idxs = seq(1, 1000, 101),
  sigma = 0.05
)
D <- data$D # Observed matrix
L_0 <- data$L # Ground truth low-rank matrix
S_0 <- data$S # Ground truth sparse matrix
Z_0 <- data$Z # Ground truth noise matrix

# Simulate a limit of detection for each chemical in mixture
lod_info <- sim_lod(D, q = 0.1)
D_lod <- lod_info$D_tilde
lod <- lod_info$lod

# Simulate missing observations
corrupted_data <- sim_na(D_lod, perc = 0.05)
D_tilde <- corrupted_data$D_tilde

# Finish simulating LOD by imputing values < LOD with LOD/sqrt(2)
lod_root2 <- matrix(
  lod / sqrt(2),
  nrow = nrow(D_tilde),
  ncol = ncol(D_tilde), byrow = TRUE
)
lod_idxs <- which(lod_info$tilde_mask == 1)
D_tilde[lod_idxs] <- lod_root2[lod_idxs]

# Run grid search to obtain optimal r, eta parameters
# (Not shown here to save space, see vignette("pcp-quickstart")
# for full example which obtains r = 3, eta = 0.224)
r_star <- 3
eta_star <- 0.224

# Run non-convex PCP to estimate L, S from D_tilde
pcp_model <- rrmc(D_tilde, r = r_star, eta = eta_star, LOD = lod)

# Clean up sparse matrix
pcp_model$S <- hard_threshold(pcp_model$S, thresh = 0.4)

# Benchmark with PCA's attempt at recovering L
D_imputed <- impute_matrix(D_tilde, apply(D_tilde, 2, mean, na.rm = TRUE))
L_pca <- proj_rank_r(D_imputed, r = r_star)

# Evaluate PCP ground truth
data.frame(
  "Obs_rel_err" = norm(L_0 - D_imputed, "F") / norm(L_0, "F"),
  "PCA_L_rel_err" = norm(L_0 - L_pca, "F") / norm(L_0, "F"),
  "PCP_L_rel_err" = norm(L_0 - pcp_model$L, "F") / norm(L_0, "F"),
  "PCP_S_rel_err" = norm(S_0 - pcp_model$S, "F") / norm(S_0, "F"),
  "PCP_L_rank" = matrix_rank(pcp_model$L),
  "PCP_S_sparsity" = sparsity(pcp_model$S)
)

References

Gibson, Elizabeth A., Junhui Zhang, Jingkai Yan, Lawrence Chillrud, Jaime Benavides, Yanelli Nunez, Julie B. Herbstman, Jeff Goldsmith, John Wright, and Marianthi-Anna Kioumourtzoglou. "Principal component pursuit for pattern identification in environmental mixtures." Environmental Health Perspectives 130, no. 11 (2022): 117008.

Tao, Rachel H., Lawrence G. Chillrud, Yanelli Nunez, Sebastian T. Rowland, Amelia K. Boehme, Jingkai Yan, Jeff Goldsmith, John Wright, and Marianthi-Anna Kioumourtzoglou. "Applying principal component pursuit to investigate the association between source-specific fine particulate matter and myocardial infarction hospitalizations in New York City." Environmental Epidemiology 7 (2), (2023).

Wu, Haotian, Vrinda Kalia, Katherine E. Manz, Lawrence Chillrud, Nathalie Hoffmann Dishon, Gabriela L. Jackson, Christian K. Dye, Raoul Orvieto, Adva Aizer, Hagai Levine, Marianthi-Anna Kioumourtzoglou, Kurt D. Pennell, Andrea A. Baccarelli, and Ronit Machtinger. "Exposome Profiling of Environmental Pollutants in Seminal Plasma and Novel Associations with Semen Parameters." Environmental Science & Technology, 58 (31), (2024): 13594-13604.

Benavides, Jaime, Sabah Usmani, Vijay Kumar, and Marianthi-Anna Kioumourtzoglou. "Development of a community severance index for urban areas in the United States: A case study in New York City." Environment International, 185, (2024): 108526.

Columbia-PRIME/pcpr documentation built on April 14, 2025, 8:33 a.m.