Function that calculates various Ridge estimators for high-dimensional precision matrices.
ridgeP(S, lambda, type = "Alt", target = default.target(S))Sample covariance matrix.
A numeric representing the value of the penalty parameter.
A character indicating the type of ridge estimator to be used. Must be one of: "Alt", "ArchI", "ArchII".
A target matrix (in precision terms) for Type I ridge estimators.
Function returns a regularized precision matrix.
The function can calculate various ridge estimators for high-dimensional precision matrices. Current (well-known) ridge estimators can be roughly divided in two archetypes. The first archetypal form employs a convex combination of \(\mathbf{S}\) and a positive definite (p.d.) target matrix \(\mathbf{T}\): \(\hat{\mathbf{\Omega}}^{\mathrm{I}}(\lambda_{\mathrm{I}}) = [(1-\lambda_{\mathrm{I}}) \mathbf{S} + \lambda_{\mathrm{I}} \mathbf{T}]^{-1}\), with \(\lambda_{\mathrm{I}} \in (0,1]\). A common target choice is for \(\mathbf{T}\) to be diagonal with \((\mathbf{T})_{jj} = (\mathbf{S})_{jj}\) for \(j=1, \ldots, p\). The second archetypal form can be given as \(\hat{\mathbf{\Omega}}^{\mathrm{II}}(\lambda_{\mathrm{II}}) = (\mathbf{S} + \lambda_{\mathrm{II}} \mathbf{I}_{p})^{-1}\) with \(\lambda_{\mathrm{II}} \in (0, \infty)\). Viewed from a penalized estimation perspective, the two archetypes utilize penalties that do not coincide with the matrix-analogue of the common ridge penalty. van Wieringen and Peeters (2015) derive analytic expressions for alternative Type I and Type II ridge precision estimators based on a proper L2-penalty. Their alternative Type I estimator (target shrinkage) takes the form $$\hat{\mathbf{\Omega}}^{\mathrm{I}a}(\lambda_{a}) = \left\{\left[\lambda_{a}\mathbf{I}_{p} + \frac{1}{4}(\mathbf{S} - \lambda_{a}\mathbf{T})^{2}\right]^{1/2} + \frac{1}{2}(\mathbf{S} - \lambda_{a}\mathbf{T})\right\}^{-1},$$ while their alternative Type II estimator can be given as a special case of the former: $$\hat{\mathbf{\Omega}}^{\mathrm{II}a}(\lambda_{a}) = \left\{\left[\lambda_{a}\mathbf{I}_{p} + \frac{1}{4}\mathbf{S}^{2}\right]^{1/2} + \frac{1}{2}\mathbf{S}\right\}^{-1}.$$ These alternative estimators were shown to be superior to the archetypes in terms of risk under various loss functions (van Wieringen and Peeters, 2015).
The lambda parameter in ridgeP generically indicates the penalty parameter. It must be chosen in accordance with the type of
ridge estimator employed. The domains for the penalty parameter in the archetypal estimators are given above. The domain for lambda in
the alternative estimators is \((0, \infty)\). The type parameter specifies the type of ridge estimator. Specifying type = "ArchI" leads
to usage of the archetypal I estimator while specifying type = "ArchII" leads to usage of the archetypal II estimator.
In the latter situation the argument target remains unused. Specifying type = "Alt"
enables usage of the alternative ridge estimators: when type = "Alt" and the target matrix is p.d. one obtains the alternative Type I estimator;
when type = "Alt" and the target matrix is specified to be the null-matrix one obtains the alternative Type II estimator.
The Type I estimators thus employ target shrinkage. The default target for both the archetype and alternative is default.target(S). When target is
not the null-matrix it is expected to be p.d. for the alternative type I estimator. The target is always expected to be p.d. in case of the archetypal I estimator.
The archetypal Type I ridge estimator is rotation equivariant when the target is of the form \(\mu\mathbf{I}_{p}\) with \(\mu \in (0,\infty)\). The archetypal
Type II estimator is rotation equivariant by definition. When the target is of the form \(\varphi\mathbf{I}_{p}\) with \(\varphi \in [0,\infty)\),
then the alternative ridge estimator is rotation equivariant. Its analytic computation is then particularly speedy as the (relatively) expensive matrix square root can then be circumvented.
van Wieringen, W.N. & Peeters, C.F.W. (2016). Ridge Estimation of Inverse Covariance Matrices from High-Dimensional Data, Computational Statistics & Data Analysis, vol. 103: 284-303. Also available as arXiv:1403.0904v3 [stat.ME].
van Wieringen, W.N. & Peeters, C.F.W. (2015). Application of a New Ridge Estimator of the Inverse Covariance Matrix to the Reconstruction of Gene-Gene Interaction Networks. In: di Serio, C., Lio, P., Nonis, A., and Tagliaferri, R. (Eds.) `Computational Intelligence Methods for Bioinformatics and Biostatistics'. Lecture Notes in Computer Science, vol. 8623. Springer, pp. 170-179.
# NOT RUN {
## Obtain some (high-dimensional) data
p = 25
n = 10
set.seed(333)
X = matrix(rnorm(n*p), nrow = n, ncol = p)
colnames(X)[1:25] = letters[1:25]
Cx <- covML(X)
## Obtain regularized precision matrix
ridgeP(Cx, lambda = 10, type = "Alt")
# }
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