sigPCA

The sigPCA package provides tools to assess the statistical significance of principal components using methods from random matrix theory, particularly the Marchenko–Pastur (MP) distribution. It also includes an optional permutation-based method for empirical validation.

Installation

# 
install.packages("pak")
pak::pak("guillermodeandajauregui/sigPCA")

Example 1: White Noise (No Signal Expected)

set.seed(123)
X_white <- matrix(rnorm(1000), nrow = 100, ncol = 10)
result_white <- sigPCA(X_white, method = "both", num_permutations = 100)
result_white$mp$significant_components
#> integer(0)
plot_sigPCA(result_white$mp$eigenvalues, result_white$mp$mp_bounds)

Example 2: Latent Structure (Signal Expected)

set.seed(456)
{
  n <- 100
  p <- 10
  k <- 2
  latent <- matrix(rnorm(n * k, mean = 3), nrow = n, ncol = k)
  loadings <- matrix(rnorm(p * k), nrow = k, ncol = p)
  noise <- matrix(rnorm(n * p, sd = 0.3), nrow = n, ncol = p)
  X_signal <- latent %*% loadings + noise
  result_signal <- sigPCA(X_signal, method = "both", num_permutations = 100)
}
result_signal$mp$significant_components
#> [1] 1 2
plot_sigPCA(result_signal$mp$eigenvalues, result_signal$mp$mp_bounds)

Interpretation

Components with eigenvalues beyond the theoretical Marchenko–Pastur upper bound are considered statistically significant. This method is particularly effective for high-dimensional datasets where traditional heuristics like scree plots may be misleading.

The permutation-based method provides empirical p-values and is useful as a secondary or confirmatory approach.

References

• Marchenko, V.A. and Pastur, L.A. (1967). Distribution of eigenvalues for some sets of random matrices. Mathematics of the USSR-Sbornik, 1(4), 457–483.
• Hernández-Lemus, E. (in preparation).