This function selects an optimal shrinkage parameter \(\rho\) for the residual covariance estimation procedure. It does so by dividing the data into groups and comparing a shrunk covariance matrix (computed on one subsample) to a benchmark covariance (computed on another subsample) using the Frobenius norm. The candidate \(\rho\) that minimizes the total squared Frobenius norm difference is selected.
adaptive_poet_rho(
R,
M0 = 10,
rho_grid = seq(0.001, 2, length.out = 20),
epsilon2 = 1e-06
)A list containing:
best_rho: The selected optimal shrinkage parameter \(\hat{\rho}\) that minimizes the total
squared Frobenius norm difference.
rho_1: The lower bound for \(\rho\) derived from the minimum eigenvalue criteria (adjusted by epsilon2).
min_Fnorm: The minimum total squared Frobenius norm difference achieved.
A numeric matrix of data (e.g., residuals) with dimensions \(T × p\), where \(T\) is the number of observations and \(p\) is the number of variables.
Integer. The number of observations to leave out between two subsamples when forming groups. Default is 10.
A numeric vector of candidate shrinkage parameters \(\rho\). Default is
seq(0.001, 2, length.out = 20).
A small positive tuning parameter used as an adjustment in the selection of \(\rho\).
Default is 1e-6.
The function proceeds as follows:
The total number of observations \(T\) is halved to define \(T_1\) and \(T_2\). Specifically: $$T_1 = \left\lfloor \frac{T}{2} \times \left(1 - \frac{1}{\log(T)}\right) \right\rfloor$$ $$T_2 = \left\lfloor \frac{T}{2} \right\rfloor - T_1$$
The sample is divided into \(\left\lfloor T/(2M_0) \right\rfloor\) groups (with \(M_0\) observations left out in between).
For each group, two subsamples are defined:
Subsample 1: the first \(T_1\) observations of the group.
Subsample 2: the last \(T_2\) observations of the group, after skipping \(M_0\) observations following subsample 1.
For each group and a given candidate \(\rho\) in rho_grid, the covariance matrix \(S_1\)
is computed from subsample 1, and then shrunk using soft-thresholding:
$$S_{1,\text{shrunk}} = \text{soft\_threshold}\left(S_1, \rho \times \text{mean}\left(|S_1|_{\text{off-diag}}\right)\right)$$
The total squared Frobenius norm between \(S_{1,\text{shrunk}}\) and the covariance matrix \(S_2\) (from subsample 2) is computed across all groups.
The function scans rho_grid to find the \(\rho\) minimizing total error. Additionally, it computes \(\rho_1\)
as \(\epsilon_2\) plus the smallest \(\rho\) for which the smallest eigenvalue of the shrunk covariance is positive.