Shrinkage Estimates of Covariance and Correlation
The functions var.shrink, cor.shrink, and cov.shrink compute
shrinkage estimates of variance, correlation, and covariance, respectively.
var.shrink(x, lambda.var, w, verbose=TRUE) cor.shrink(x, lambda, w, verbose=TRUE) cov.shrink(x, lambda, lambda.var, w, verbose=TRUE)
x |
a data matrix |
lambda |
the correlation shrinkage intensity (range 0-1).
If |
lambda.var |
the variance shrinkage intensity (range 0-1).
If |
w |
optional: weights for each data point - if not specified uniform weights are assumed
( |
verbose |
output some status messages while computing (default: TRUE) |
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).
they are typically much more efficient, i.e. they show (sometimes dramatically) better mean squared error,
the estimated covariance and correlation matrices are always positive definite and well conditioned (so that there are no numerical problems when computing their inverse),
they are inexpensive to compute, and
they are fully automatic and do not require any tuning parameters (as the shrinkage intensity is analytically estimated from the data), and
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.
var.shrink returns a vector with estimated variances.
cov.shrink returns a covariance matrix.
cor.shrink returns the corresponding correlation matrix.
Juliane Sch\"afer, Rainer Opgen-Rhein, 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>
# 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))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.