shrink.intensity: Estimation of Shrinkage Intensities
In corpcor: Efficient Estimation of Covariance and (Partial) Correlation

Description Usage Arguments Details Value Author(s) References See Also Examples

The functions estimate.lambda and estimate.lambda.var shrinkage intensities used for correlations and variances used in cor.shrink and var.shrink, respectively.

1 2	estimate.lambda(x, w, verbose=TRUE) estimate.lambda.var(x, w, verbose=TRUE)

`x`	a data matrix
`w`	optional: weights for each data point - if not specified uniform weights are assumed (`w = rep(1/n, n)` with `n = nrow(x)`).
`verbose`	report shrinkage intensities (default: TRUE)

var.shrink computes the empirical variance of each considered random variable, and shrinks them towards their median. The corresponding shrinkage intensity lambda.var is estimated using

λ_{var}^{*} = ( ∑_{k=1}^p Var(s_{kk}) )/ ∑_{k=1}^p (s_{kk} - median(s))^2

where median(s) denotes the median of the empirical variances (see Opgen-Rhein and Strimmer 2007).

Similarly, cor.shrink computes a shrinkage estimate of the correlation matrix by shrinking the empirical correlations towards the identity matrix. In this case the shrinkage intensity lambda equals

λ^{*} = ∑_{k \neq l} Var(r_{kl}) / ∑_{k \neq l} r_{kl}^2

(Sch\"afer and Strimmer 2005).

Ahdesm\"aki suggested (2012) a computationally highly efficient algorithm to compute the shrinkage intensity estimate for the correlation matrix (see the R code for the implementation).

estimate.lambda and estimate.lambda.var returns a number between 0 and 1.

Juliane Sch\"afer, Rainer Opgen-Rhein, Miika Ahdesm\"aki and Korbinian Strimmer (https://strimmerlab.github.io).

Opgen-Rhein, R., and K. Strimmer. 2007. Accurate ranking of differentially expressed genes by a distribution-free shrinkage approach. Statist. Appl. Genet. Mol. Biol. 6:9. <DOI:10.2202/1544-6115.1252>

Sch\"afer, J., and K. Strimmer. 2005. A shrinkage approach to large-scale covariance estimation and implications for functional genomics. Statist. Appl. Genet. Mol. Biol. 4:32. <DOI:10.2202/1544-6115.1175>

cor.shrink, var.shrink.

# load corpcor library
library("corpcor")

# small n, large p
p = 100
n = 20

# generate random pxp covariance matrix
sigma = matrix(rnorm(p*p),ncol=p)
sigma = crossprod(sigma)+ diag(rep(0.1, p))

# simulate multinormal data of sample size n  
sigsvd = svd(sigma)
Y = t(sigsvd$v %*% (t(sigsvd$u) * sqrt(sigsvd$d)))
X = matrix(rnorm(n * ncol(sigma)), nrow = n) %*% Y


# correlation shrinkage intensity
estimate.lambda(X) 
c = cor.shrink(X)
attr(c, "lambda")

# variance shrinkage intensity
estimate.lambda.var(X) 
v = var.shrink(X)
attr(v, "lambda.var")