Internal Documentation

epanechnikov(x)

Return the Epanechnikov kernel evaluated at x.

epnanechnikov_HT(x)

Return the Hilbert Transform of the Epanechnikov kernel evaluated at x if |x|≂̸√5.

epnanechnikov_HT2(x)

Return the Hilbert Transform of the Epanechnikov kernel evaluated at x if |x|=√5.

rescale(M, d)

Internal function to scale the rows and the columns of a square matrix M according to the elements in d. This is useful when dealing with the standardised data matrix which can be written Xs=Xc*D where Xc is the centered data matrix and D=Diagonal(d) so that its simple covariance is D*S*D where S is the simple covariance of Xc. Such D*M*D terms appear often in the computations of optimal shrinkage λ.

• Space complexity: $O(p^2)$
• Time complexity: $O(p^2)$

rescale!(M, d)

Same as rescale but in place (no allocation).

uccov(X)

Internal function to compute X*X'/n where n is the number of rows of X. This corresponds to the uncorrected covariance of X if X is centered. This operation appears often in the computations of optimal shrinkage λ.

• Space complexity: $O(p^2)$
• Time complexity: $O(2np^2)$

sumij(S)

Internal function to compute the sum of elements of a square matrix S. A keyword with_diag can be passed to indicate whether to include or not the diagonal of S in the sum. Both cases happen often in the computations of optimal shrinkage λ.

• Space complexity: $O(1)$
• Time complexity: $O(p^2)$

sumij2(S)

Internal function identical to sumij except that it passes the function abs2 to the sum so that it is the sum of the elements of S squared which is computed. This is significantly more efficient than using sumij(S.^2) for large matrices as it allocates very little.

• Space complexity: $O(1)$
• Time complexity: $O(2p^2)$

sum_fij(Xc, S, n, κ)

Internal function corresponding to $∑_{i≂̸j}f_{ij}$ that appears in https://strimmerlab.github.io/publications/journals/shrinkcov2005.pdf p.11.

• Space complexity: $O(np + 2p^2)$
• Time complexity: $O(2np^2)$

linear_shrinkage(target, Xc, S, λ, n, p, pvar, corrected, [weights])

Performs linear shrinkage with target of type target for data matrix Xc of size n by p with covariance matrix S and shrinkage parameter λ. Calculates corrected covariance if corrected is true.

pvar == p or pvar = p - sum(iszero, diag(S)), the number of non-zero diagonal variances in S. The choice is controlled by LinearShrinkage(...; drop_var0=true/false).

analytical_nonlinear_shrinkage(S, n, p; decomp;
    alpha::Real=0.0,
    warn_if_small_eigenvalue::Bool=false,
)

Internal implementation of the analytical nonlinear shrinkage. The implementation is inspired from the Matlab code given in section C of Olivier Ledoit and Michael Wolf's paper "Analytical Nonlinear Shrinkage of Large-Dimensional Covariance Matrices". (Nov 2018) http://www.econ.uzh.ch/static/wp/econwp264.pdf , later published as

O. Ledoit and M. Wolf, “Analytical nonlinear shrinkage of large-dimensional covariance
matrices,” The Annals of Statistics, vol. 48, no. 5, pp. 3043–3065, Oct. 2020,
doi: 10.1214/19-AOS1921.

Shrinked eigenvalues smaller than alpha are replaced with alpha. If warn_if_small_eigenvalue is true, additionally a warning is emitted if any eigenvalue is replaced with alpha.