corpcor (version 1.2.0)

cov.shrink: Shrinkage Estimates of Covariance and Correlation

Description

The functions cov.shrink and cor.shrink compute shrinkage estimates of covariance and correlation, as as suggested in Schaefer and Strimmer (2005). In comparison with the standard empirical estimates (cov and cor) the shrinkage estimates have a number of advantages

1. are always positive definite,
2. well conditioned (so that the inverse always exists), and
3. exhibit (sometimes dramatically) better mean squared error.

Furthermore, they are inexpensive to compute and do not require any tuning parameters (the shrinkage intensity is analytically estimated from the data). As an extra benefit, the shrinkage estimators have a form that can be *very* efficiently inverted using the Woodbury matrix identity, even if the number of variables is large. This approach is implemented 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.

Usage

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

Arguments

x

a data matrix

lambda

the shrinkage intensity (range 0-1). If $\lambda$ is is not specified (the default) a suitable value is automatically chosen such that the resulting shrinkage estimate has minimal MSE (see below for details). For $\lambda=0$ not shrinkage occur

w

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

verbose

output status while computing (default: TRUE)

Value

• cov.shrink returns a covariance matrix. cor.shrink returns the corresponding correlation matrix.

Details

cor.shrink computes a shrinkage estimate $R^{*}$ of the correlation matrix according to $$R^{*} = \lambda T + (1-\lambda) R$$ where $R$ is the usual empirical correlation matrix and the target $T$ is the unit diagonal matrix. The shrinkage intensity $\lambda^{*}$ for which the MSE of $R^{*}$ is minimal is estimated by $$\lambda^{*} = \sum_{i \neq j} Var(r_{ij}) / \sum_{i \neq j} r_{ij}^2 .$$ This is a special case of the analytic formula by Ledoit and Wolf (2003) for the optimal shrinkage (note that in contrast to Ledoit-Wolf here it is applied to the correlation rather than the covariance matrix).

cov.shrink computes the corresponding full covariance matrix on the basis of the shrunken correlation matrix and the empirical variances. These shrinkage estimator are especially useful in a 'small n, large p' setting - this is often encountered, e.g., in genomics. For an extensive discussion please see Schaefer and Strimmer (2005).

References

Ledoit, O., and Wolf. M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. J. Emp. Finance 10:503-621.

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))