corpcor (version 1.6.10)

cov.shrink: Shrinkage Estimates of Covariance and Correlation

Description

The functions var.shrink, cor.shrink, and cov.shrink compute shrinkage estimates of variance, correlation, and covariance, respectively.

Usage

var.shrink(x, lambda.var, w, verbose=TRUE)
cor.shrink(x, lambda, w, verbose=TRUE)
cov.shrink(x, lambda, lambda.var, w, verbose=TRUE)

Arguments

x

a data matrix

lambda

the correlation shrinkage intensity (range 0-1). If lambda is not specified (the default) it is estimated using an analytic formula from Sch\"afer and Strimmer (2005) - see details below. For lambda=0 the empirical correlations are recovered.

lambda.var

the variance shrinkage intensity (range 0-1). If lambda.var is not specified (the default) it is estimated using an analytic formula from Opgen-Rhein and Strimmer (2007) - see details below. For lambda.var=0 the empirical variances are recovered.

w

optional: weights for each data point - if not specified uniform weights are assumed (w = rep(1/n, n) with n = nrow(x)).

verbose

output some status messages while computing (default: TRUE)

Value

var.shrink returns a vector with estimated variances.

cov.shrink returns a covariance matrix.

cor.shrink returns the corresponding correlation matrix.

Details

var.shrink computes the empirical variance of each considered random variable, and shrinks them towards their median. The shrinkage intensity is estimated using estimate.lambda.var (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 is computed using estimate.lambda (Sch\"afer and Strimmer 2005).

In comparison with the standard empirical estimates (var, cov, and cor) the shrinkage estimates exhibit a number of favorable properties. For instance,

1. they are typically much more efficient, i.e. they show (sometimes dramatically) better mean squared error,

2. the estimated covariance and correlation matrices are always positive definite and well conditioned (so that there are no numerical problems when computing their inverse),

3. they are inexpensive to compute, and

4. they are fully automatic and do not require any tuning parameters (as the shrinkage intensity is analytically estimated from the data), and

5. they assume nothing about the underlying distributions, except for the existence of the first two moments.

These properties also carry over to derived quantities, such as partial variances and partial correlations (pvar.shrink and pcor.shrink).

As an extra benefit, the shrinkage estimators have a form that can be very efficiently inverted, especially if the number of variables is large and the sample size is small. Thus, instead of inverting the matrix output by cov.shrink and cor.shrink please use the functions invcov.shrink and invcor.shrink, respectively.

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

invcov.shrink, pcor.shrink, cor2pcor

Examples

# NOT RUN {
# 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


# estimate covariance matrix
s1 = cov(X)
s2 = cov.shrink(X)


# squared error
sum((s1-sigma)^2)
sum((s2-sigma)^2)


# compare positive definiteness
is.positive.definite(sigma)
is.positive.definite(s1)
is.positive.definite(s2)


# compare ranks and condition
rank.condition(sigma)
rank.condition(s1)
rank.condition(s2)

# compare eigenvalues
e0 = eigen(sigma, symmetric=TRUE)$values
e1 = eigen(s1, symmetric=TRUE)$values
e2 = eigen(s2, symmetric=TRUE)$values
m = max(e0, e1, e2)
yl = c(0, m)

par(mfrow=c(1,3))
plot(e1,  main="empirical")
plot(e2,  ylim=yl, main="full shrinkage")
plot(e0,  ylim=yl, main="true")
par(mfrow=c(1,1))

# }