corpcor (version 1.4.0)

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 Schaefer and Strimmer (2005) - see details below. For lambda=0 the

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 Schaefer and Strimmer (2005) - see details below. F

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 common average. The shrinkage intensity is estimated using Eq. 6 from Schaefer and Strimmer (2005), which leads to $$\lambda_{var}^{*} = ( (1-1/p) \sum_{k=1}^p Var(s_{kk}) )/ \sum_{k=1}^p (s_{kk} - \bar{s})^2 .$$

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 $$\lambda^{*} = \sum_{k \neq k} Var(r_{kl}) / \sum_{k \neq l} r_{kl}^2 .$$

cov.shrink computes the corresponding full covariance matrix on the basis of the shrunken correlation matrix and the shrunken variances. See also Opgen-Rhein and Strimmer (2006). 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 covariances and partial correlations (pcov.shrink and pcor.shrink).

As an extra benefit, the shrinkage estimators have a form that can be very efficiently inverted using the Woodbury matrix identity, especially if the number of variables is large and the sample size is small. Note that this identity is employed in the functions \code{invcov.shrink} and invcor.shrink. and is much faster than directly inverting the matrix output by cov.shrink and cor.shrink, respectively.

References

Opgen-Rhein, R., and Strimmer, K. (2006). Beyond Gauss-Markov and ridge: an effective approach for high-dimensional regression and causal inference. Submitted.

Schaefer, J., and Strimmer, K. (2005). A shrinkage approach to large-scale covariance estimation and implications for functional genomics. Statist. Appl. Genet. Mol. Biol.4:32. (http://www.bepress.com/sagmb/vol4/iss1/art32/)

See Also

invcov.shrink, pcor.shrink, cor2pcor

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


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

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

# compare eigenvalues
e1 <- eigen(s1, symmetric=TRUE)$values
e2 <- eigen(s2, symmetric=TRUE)$values
e3 <- eigen(sigma, symmetric=TRUE)$values
m <-max(e1, e2, e3)
yl <- c(0, m)

par(mfrow=c(1,3))
plot(e1,  main="empirical")
plot(e2,  ylim=yl, main="shrinkage")
plot(e3,  ylim=yl, main="true")
par(mfrow=c(1,1))