adaptive_poet_rho {TVMVP} | R Documentation |
Adaptive Selection of the Shrinkage Parameter \rho
for POET
Description
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.
Usage
adaptive_poet_rho(
R,
M0 = 10,
rho_grid = seq(0.001, 2, length.out = 20),
epsilon2 = 1e-06
)
Arguments
R |
A numeric matrix of data (e.g., residuals) with dimensions |
M0 |
Integer. The number of observations to leave out between two subsamples when forming groups. Default is 10. |
rho_grid |
A numeric vector of candidate shrinkage parameters |
epsilon2 |
A small positive tuning parameter used as an adjustment in the selection of |
Details
The function proceeds as follows:
The total number of observations
T
is halved to defineT_1
andT_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 (withM_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 skippingM_0
observations following subsample 1.
For each group and a given candidate
\rho
inrho_grid
, the covariance matrixS_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 matrixS_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.
Value
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 byepsilon2
). -
min_Fnorm
: The minimum total squared Frobenius norm difference achieved.