shrink.intensity {corpcor}

Estimation of Shrinkage Intensities

Description

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

Usage

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

Arguments

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)

Details

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

\lambda_{var}^{*} = ( \sum_{k=1}^p Var(s_{kk}) )/ \sum_{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

\lambda^{*} = \sum_{k \neq l} Var(r_{kl}) / \sum_{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).

Value

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

Author(s)

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

References

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>

See Also

cor.shrink, var.shrink.

Examples

# 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")