robustPoetEst() implements the robust version of
Principal Orthogonal complEment Thresholding (POET) estimator, a
nonparametric, unobserved-factor-based estimator of the covariance matrix
when the underlying distribution is elliptical
(Fan et al. 2018)
. The estimator is defined as the sum of the
sample covariance matrix's rank-k approximation and its
post-thresholding principal orthogonal complement. The rank-k
approximation is constructed from the sample covariance matrix, its leading
eigenvalues, and its leading eigenvectors. The sample covariance matrix and
leading eigenvalues are initially estimated via an M-estimation procedure
and the marginal Kendall's tau estimator. The leading eigenvectors are
estimated using spatial Kendall's tau estimator. The hard thresholding
function is used to regularize the idiosyncratic errors' estimated
covariance matrix, though other regularization schemes could be used.
We do not recommend that this estimator be employed when the estimand is the correlation matrix. The diagonal entries of the resulting estimate are not guaranteed to be equal to one.
robustPoetEst(dat, k, lambda, var_est = c("sample", "mad", "huber"))A numeric data.frame, matrix, or similar object.
An integer indicating the number of unobserved latent
factors. Empirical evidence suggests that the POET estimator is robust to
overestimation of this hyperparameter (Fan et al. 2013)
. In
practice, it is therefore preferable to use larger values.
A non-negative numeric defining the amount of
thresholding applied to each element of sample covariance matrix's
orthogonal complement.
A character dictating which variance estimator to
use. This must be one of the strings "sample", "mad", or
"huber". "sample" uses sample variances; "mad"
estimates variances via median absolute deviation; "huber" uses an
M-estimator for variance under the Huber loss.
A matrix corresponding to the estimate of the covariance
matrix.
Fan J, Liao Y, Mincheva M (2013).
“Large covariance estimation by thresholding principal orthogonal complements.”
Journal of the Royal Statistical Society. Series B (Statistical Methodology), 75(4), 603--680.
ISSN 13697412, 14679868, https://www.jstor.org/stable/24772450.
Fan J, Liu H, Wang W (2018).
“Large covariance estimation through elliptical factor models.”
Ann. Statist., 46(4), 1383--1414.
doi: 10.1214/17-AOS1588
.
robustPoetEst(dat = mtcars, k = 2L, lambda = 0.1, var_est = "sample")
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 36.315178 -9.1164387 -693.67785 -376.03067 2.11448003 -5.3349120
#> [2,] -9.116439 2.7537461 190.39219 102.45344 -0.61276404 1.3986029
#> [3,] -693.677853 190.3921862 15358.02683 7267.79320 -46.11029191 110.9814745
#> [4,] -376.030665 102.4534357 7267.79320 4696.88357 -20.27007620 54.3234998
#> [5,] 2.114480 -0.6127640 -46.11029 -20.27008 0.21013981 -0.4787939
#> [6,] -5.334912 1.3986029 110.98147 54.32350 -0.47879387 0.9573022
#> [7,] 5.076855 -1.7307761 -100.04564 -81.84586 0.04839227 -0.3844377
#> [8,] 1.851765 -0.6218331 -38.85915 -22.17540 0.15703000 -0.2555406
#> [9,] 1.484410 -0.3848420 -33.54033 -11.16801 0.20367457 -0.3061374
#> [10,] 2.280022 -0.6167975 -51.04325 -17.11729 0.18503478 -0.4043015
#> [11,] -6.199445 1.4396723 107.50032 80.26554 -0.11182020 0.7713457
#> [,7] [,8] [,9] [,10] [,11]
#> [1,] 5.07685520 1.85176484 1.48440950 2.2800219 -6.19944529
#> [2,] -1.73077606 -0.62183308 -0.38484203 -0.6167975 1.43967234
#> [3,] -100.04564404 -38.85915432 -33.54032586 -51.0432471 107.50032232
#> [4,] -81.84586275 -22.17539543 -11.16800914 -17.1172950 80.26554184
#> [5,] 0.04839227 0.15703000 0.20367457 0.1850348 -0.11182020
#> [6,] -0.38443765 -0.25554058 -0.30613738 -0.4043015 0.77134565
#> [7,] 3.19310208 0.60378091 -0.16563939 -0.1499862 -1.85214514
#> [8,] 0.60378091 0.11854837 0.03335364 0.1292760 -0.37857863
#> [9,] -0.16563939 0.03335364 0.21142802 0.2100142 -0.04591921
#> [10,] -0.14998618 0.12927602 0.21001418 0.4573464 0.12806927
#> [11,] -1.85214514 -0.37857863 -0.04591921 0.1280693 2.44068633