Calculate confidence interval for log determinant of covariance matrices

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Description

This function uses asymptotic results described by Cai et. al (2016), Theorem 1, to calculate approximate, normal theory confidence intervals (CIs) for the log determinant of one or more sample covariance matrices.

Their results are translated into a CI via the approximation

\log det( \widehat{Σ} ) - bias \pm z_{1 - α/2} \times SE

where \widehat{Σ} is the sample estimate of a population covariance matrix, bias is a bias correction constant and SE is a width factor for the confidence interval. Both bias and SE are functions of the sample size, n and number of variables, p.

This function is included here only to provide an approximation to graphical accuracy for use with Box's M test for equality of covariance matrices, boxM and its associated plot.boxM method.

Usage

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logdetCI(cov, n, conf = 0.95, method = 1, bias.adj = TRUE)

Arguments

cov

a covariance matrix or a (named) list of covariance matrices, all the same size

n

sample size, or vector of sample sizes, one for each covariance matrix

conf

confidence level

method

Three methods are provided, based on Cai et. al Theorem 1 (method=1), Colollary 1 (method=2) and Colollary 2 (method=3), each with different bias and SE values.

bias.adj

logical; set FALSE to exclude the bias correction term.

Details

Cai et. al (2016) claim that their Theorem 1 holds with either p fixed or p(n) growing with n, as long as p(n) ≤ n. Their Corollary 1 (method=2) is the special case when p is fixed. Their Corollary 2 (method=3) is the special case when 0 ≤ p/n < 1 is fixed.

The properties of this CI estimator are unknown in small to moderate sample sizes, but it seems to be the only one available. It is therefore experimental in this version of the package and is subject to change in the future.

The bias term offsets the confidence interval from the sample estimate of \log det( \widehat{Σ} ) . When p is large relative to n, the confidence interval may not overlap the sample estimate.

Strictly speaking, this estimator applies to the MLE of the covariance matrix \widehat{Σ}, i.e., using n rather than n-1 in as the divisor. The factor (n-1 / n) has not yet been taken into account here.

Value

A data frame with one row for each covariance matrix. lower and upper are the boundaries of the confidence intervals. Other columns are logdet, bias, se.

Author(s)

Michael Friendly

References

Cai, T. T., Liang, T. & Zhou, H. H. (2016). Law of log determinant of sample covariance matrix and optimal estimation of differential enteropy for high-dimensional Gaussian distributions. JMA, to appear. Preprint: www.stat.yale.edu/~hz68/Covariance-Determinant.pdf

See Also

boxM, plot.boxM

Examples

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data(iris)
iris.mod <- lm(as.matrix(iris[,1:4]) ~ iris$Species)
iris.boxm <- boxM(iris.mod)
cov <- c(iris.boxm$cov, list(pooled=iris.boxm$pooled))
n <- c(rep(50, 3), 150)

CI <- logdetCI( cov, n=n, conf=.95, method=1)
CI
plot(iris.boxm, xlim=c(-14, -8), main="Iris data, Box's M test", gplabel="Species")
arrows(CI$lower, 1:4, CI$upper, 1:4, lwd=3, angle=90, len=.1, code=3)

CI <- logdetCI( cov, n=n, conf=.95, method=1, bias.adj=FALSE)
CI
plot(iris.boxm, xlim=c(-14, -8), main="Iris data, Box's M test", gplabel="Species")
arrows(CI$lower, 1:4, CI$upper, 1:4, lwd=3, angle=90, len=.1, code=3)

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