Quickstart: applying PCP to a simulated environmental mixture

Simulating data

The sim_data() function lets users generate simple mixtures models for quick experimentation. Let’s use its default parameters to simulate a noisy environmental mixture \(D = L_0 + S_0 + Z_0\) comprised of \(n = 100\) observations of \(p = 10\) chemicals, with three underlying chemical exposure patterns (or a rank \(r = 3\)), extreme outlying exposure events along the diagonal of the matrix, and dense Gaussian noise corrupting all the measurements in the matrix:

data <- sim_data()
D <- data$D
L_0 <- data$L
S_0 <- data$S
Z_0 <- data$Z

Let’s get a sense of what these matrices look like. Remember in real world analyses, we only observe \(D\). The matrices \(L_0\), \(S_0\), and \(Z_0\) are the ground truth, coming from some unobserved data generating mechanism:

plot_matrix(D)

plot_matrix(L_0)

plot_matrix(S_0)

plot_matrix(Z_0)

The matrix_rank() function estimates the rank of a given matrix by counting the number of nonzero singular values governing that matrix (with the help of a thresh parameter determining what is “practically zero”):

matrix_rank(L_0)
#> [1] 3
matrix_rank(D)
#> [1] 10

Here we can see L_0 has 3 underlying patterns. This is obscured in the observed, full-rank mixture matrix D, since the S_0 and Z_0 components are corrupting the underlying structure provided by L_0.

Let’s simulate the chemicals (columns) of our data being subject to some limit of detection (LOD) with the sim_lod() function. We will simulate the bottom 10th percentile of each column as being below LOD, and examine the LOD for each column:

lod_info <- sim_lod(D, q = 0.1)
D_lod <- lod_info$D_tilde
lod <- lod_info$lod
lod
#>  [1] 0.2573431 0.5325583 0.2053927 0.2329695 0.1620019 0.3801870 0.6461348
#>  [8] 0.1499115 0.5306228 0.6183821

Next, because our mixtures data is often incomplete in practice, let’s further simulate a random 5% of the values as missing NA with the sim_na() function. Once our missing values are in place, we can finish simulating the LOD in the mixture by imputing values simulated < LOD to be the \(LOD / \sqrt{2}\), a common imputation scheme for measurements < LOD:

corrupted_data <- sim_na(D_lod, perc = 0.05)
D_tilde <- corrupted_data$D_tilde
lod_root2 <- matrix(
  lod / sqrt(2),
  nrow = nrow(D_tilde),
  ncol = ncol(D_tilde), byrow = TRUE
)
lod_idxs <- which(lod_info$tilde_mask == 1)
D_tilde[lod_idxs] <- lod_root2[lod_idxs]
plot_matrix(D_tilde)

The D_tilde matrix represents our observed, messy mixtures model, suffering from incomplete NA observations and a chemical-specific LOD.

Model selection

There are two PCP algorithms shipped with pcpr: the convex root_pcp() [Zhang et al. (2021)] and non-convex rrmc() [Cherapanamjeri et al. (2017)]. To figure out which model would be best for our data, let’s inspect the singular values of our observed mixture using the sing() method (sing() cannot accept missing values, so we will use impute_matrix() to impute NA values in D_tilde with their respective column means):

D_imputed <- impute_matrix(D_tilde, apply(D_tilde, 2, mean, na.rm = TRUE))
singular_values <- sing(D_imputed)
plot(singular_values, type = "b")

rrmc() is best suited for data characterized by slowly decaying singular values, indicative of complex underlying patterns and a relatively large degree of noise. Most EH data can be described this way. root_pcp() is best for data characterized by rapidly decaying singular values, indicative of very well-defined latent patterns.

The singular values plotted above decay quickly from the first to the second, but very gradually from the second onward. For this simple simulated dataset, both PCP models are perfectly suitable. We will use rrmc(), as this is the model environmental health researchers will likely employ most frequently. The vignette("pcp-applied") contains an exemplary mixtures matrix singular value plot with slowly decaying singular values.

Grid search for parameter tuning

To estimate the low-rank and sparse matrices, rrmc() needs to be given a maximum rank r and regularization parameter eta. To determine the optimal values of r and eta, we will conduct a brief grid search using the grid_search_cv() function. In our grid search below, we will examine all models from ranks 1 through 5 and all values of eta near the default, calculated with get_pcp_defaults().

eta_0 <- get_pcp_defaults(D_tilde)$eta
etas <- data.frame("eta" = sort(c(0.1 * eta_0, eta_0 * seq(1, 10, 2))))
# to get progress bar, could wrap this
# in a call to progressr::with_progress({ gs <- grid_search_cv(...) })
gs <- grid_search_cv(
  D_tilde,
  pcp_fn = rrmc,
  grid = etas, r = 5, LOD = lod,
  parallel_strategy = "multisession",
  num_workers = 16,
  verbose = FALSE
)

The rrmc() approach to PCP uses an iterative rank-based procedure to recover L and S, meaning it first constructs a rank 1 model and iteratively builds up to the specified rank r solution. As such, for the above grid search, we passed etas as the grid argument to search and sent \(r = 5\) as a constant parameter common to all models in the search. Since length(etas) = 6 and \(r = 5\), we searched through 30 different PCP models. The num_runs argument determines how many (random) tests should be performed for each unique model setting. By default, num_runs = 100, so our grid search tuned r and eta by measuring the performance of 3000 different PCP models. We passed the simulated lod vector as another constant to the grid search, equipping each rrmc() run with the same LOD information.

r_star <- gs$summary_stats$r[1]
eta_star <- round(gs$summary_stats$eta[1], 3)
gs$summary_stats
#> # A tibble: 30 × 7
#>      eta     r rel_err L_rank S_sparsity iterations run_error_perc
#>    <dbl> <int>   <dbl>  <dbl>      <dbl>      <dbl> <chr>         
#>  1 0.224     3   0.145   3         0.968        NaN 0%            
#>  2 0.313     3   0.150   3         0.984        NaN 0%            
#>  3 0.224     4   0.167   4         0.967        NaN 0%            
#>  4 0.134     5   0.168   2.21      0.199        NaN 0%            
#>  5 0.134     3   0.168   2.21      0.199        NaN 0%            
#>  6 0.134     4   0.168   2.21      0.199        NaN 0%            
#>  7 0.402     3   0.169   3         0.993        NaN 0%            
#>  8 0.134     2   0.173   2         0.342        NaN 0%            
#>  9 0.224     2   0.174   2         0.992        NaN 0%            
#> 10 0.402     2   0.182   2         1.00         NaN 0%            
#> # ℹ 20 more rows

Inspecting the summary_stats table from the output grid search provides the mean-aggregated statistics for each of the 30 distinct parameter settings we tested. The grid search correctly identified the rank 3 solution as the best (lowest relative error rel_err rate). The corresponding eta = 0.224. The top three parameter settings also seem to have reasonable S_sparsity levels as well (all are above 0.95). The next three parameter settings seem to under-regularize the sparse S matrix by quite a bit, as 80% of entries are non-zero. We will take the top parameters identified by the grid search in this instance. Had the very top parameters yielded a sparsity of e.g. 0.7, we likely then would have preferred the second set of parameters with sparisities in the 0.9s. This decision would have been grounded in prior assumptions about the amount of outliers to expect in the mixtuere. For more on the interpreation of grid search results, consult the documentation for the grid_search_cv() function.

Running PCP

Now we can run our PCP model:

pcp_model <- rrmc(D_tilde, r = r_star, eta = eta_star, LOD = lod)

We can inspect the evolution of the objective function over the course of PCP’s optimization:

plot(pcp_model$objective, type = "l")

And the output L matrix:

plot_matrix(pcp_model$L)

matrix_rank(pcp_model$L)
#> [1] 3

Let’s briefly inspect PCP’s estimate of the sparse matrix S, and fix any values that are “practically” zero using the hard_threshold() function. The histogram below shows a majority of the entries in S are between -0.4 and 0.4, so we will call those values “practically” zero, and the rest true outlying exposure events. We can then calculate the sparsity of S with sparsity():

hist(pcp_model$S)

pcp_model$S <- hard_threshold(pcp_model$S, thresh = 0.4)
plot_matrix(pcp_model$S)

sparsity(pcp_model$S)
#> [1] 0.989

Benchmarking with PCA

Before evaluating our PCP model, let’s see how well a more traditional method such as Principal Component Analysis (PCA) can recover L_0, to provide a benchmark for comparison.

The proj_rank_r() function (project matrix to rank r) approximates an input matrix as low-rank using a rank-r truncated SVD, the same way PCA approximates a low-rank matrix. Normally, a researcher would need to determine r subjectively. We will give PCA an advantage by sharing PCP’s discovery from the above grid search that the solution should be of rank 3:

L_pca <- proj_rank_r(D_imputed, r = r_star)

Evaluating PCP against the ground truth

Finally, let’s see how we did in recovering L_0 and S_0. We will examine the relative error between our model’s estimates and the simulated ground truth matrices. We use the Frobenius norm to calculate the relative errors between the matrices:

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)
)
#>   Obs_rel_err PCA_L_rel_err PCP_L_rel_err PCP_S_rel_err PCP_L_rank
#> 1   0.1440249    0.08096932    0.05847706      0.232115          3
#>   PCP_S_sparsity
#> 1          0.989

PCP outperformed PCA by quite a bit! PCP’s relative recovery error on the L_0 matrix stood at only 5.85%, compared to an observed relative error of 14.4% and PCA’s relative error of 8.1%. PCP’s sparse matrix estimate was only off from the ground truth S_0 by 23.21%.

After PCP

We can now pair our estimated L matrix with any matrix factorization method of our choice (e.g. PCA, factor analysis, or non-negative matrix factorization) to extract the latent chemical exposure patterns (an example of what this looks like is in vignette("pcp-applied"), where non-negative matrix factorization is used to extract patterns from PCP’s L matrix). These patterns, along with the isolated outlying exposure events in S, can then be analyzed with any outcomes of interest in downstream epidemiological analyses.