CoFM (version 1.1.4)

poet: POET: Principal Orthogonal complEment Thresholding

Description

Implements the POET method for large covariance matrix estimation (Fan, Liao & Mincheva, 2013). The method assumes a factor model structure, estimates the low-rank component via PCA, and applies thresholding to the sparse residual covariance matrix.

Usage

poet(
  X,
  r = NULL,
  r.max = 10,
  thresh = "hard",
  lambda = NULL,
  gamma = 3.7,
  delta = 1e-04,
  method.r = "IC1"
)

Value

A list containing:

Sigma.poet

The estimated N x N POET covariance matrix.

Sigma.fact

The estimated N x N low-rank (factor) covariance matrix.

Sigma.resid

The estimated N x N thresholded residual covariance matrix.

F.hat

Estimated factors (T x r).

Lambda.hat

Estimated factor loadings (N x r).

r.hat

The number of factors used (estimated or specified).

R.hat

Same as Sigma.resid (for compatibility).

Arguments

X

Numeric matrix (T x N). T is the number of time periods (rows), N is the number of variables (columns).

r

Integer or NULL. User-specified number of factors. If NULL, r is estimated automatically using method.r.

r.max

Integer. Upper bound for the number of factors when estimating r. Default is 10.

thresh

Character. Thresholding type for the residual covariance. Options: "hard", "soft", "scad", "adapt". Default is "hard".

lambda

Numeric or NULL. Thresholding parameter. If NULL, it defaults to sqrt(log(N)/T).

gamma

Numeric. Parameter for SCAD thresholding. Default is 3.7.

delta

Numeric. Minimum eigenvalue bump to ensure positive definiteness of the residual covariance. Default is 1e-4.

method.r

Character. Method to select the number of factors if r is NULL. Options: "IC1" (Bai & Ng, 2002) or "ER" (Eigenvalue Ratio, Ahn & Horenstein, 2013).

References

Fan, J., Liao, Y., & Mincheva, M. (2013). Large covariance estimation by thresholding principal orthogonal complements. Journal of the Royal Statistical Society: Series B, 75(4), 603-680.

Examples

# Examples should be fast and reproducible for CRAN checks
set.seed(2025)
T_obs <- 40; N_var <- 15; r_true <- 2

# Generate a simple factor model: X = F * Lambda' + U
Lambda <- matrix(stats::rnorm(N_var * r_true), N_var, r_true)
F_scores <- matrix(stats::rnorm(T_obs * r_true), T_obs, r_true)
U <- matrix(stats::rnorm(T_obs * N_var), T_obs, N_var)
X_sim <- F_scores %*% t(Lambda) + U  # T x N

# Apply POET (choose r via IC1; use soft thresholding)
res <- poet(X_sim, r = NULL, method.r = "IC1", thresh = "soft")
res$r.hat
res$Sigma.poet[1:5, 1:5]