heplots (version 1.8.5)

traceCI: Calculate Confidence Interval for Trace of Covariance Matrices

Description

This function uses asymptotic results from random matrix theory to calculate approximate, normal theory confidence intervals for the trace (sum of eigenvalues) of one or more sample covariance matrices.

The trace of a covariance matrix equals the sum of its eigenvalues, which represents the total variance in the system. For large samples, the trace follows an approximate normal distribution with variance that can be calculated from the eigenvalue structure.

Usage

traceCI(cov, n, conf = 0.95)

Value

A data frame with one row for each covariance matrix. Columns:

trace

The trace (sum of eigenvalues) of the covariance matrix

se

Standard error of the trace

lower

Lower confidence limit

upper

Upper confidence limit

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 (default: 0.95)

Author

Michael Friendly

Details

The confidence interval is based on the asymptotic normality of the trace: $$ trace(\widehat{\Sigma}) \pm z_{1 - \alpha/2} \times SE $$ where \(\widehat{\Sigma}\) is the sample covariance matrix and \(SE\) is the standard error.

The variance of the trace is calculated using the formula from Bai & Silverstein (2004): $$ Var(trace) = \frac{2}{n} \sum_{i=1}^{p} \lambda_i^2 = \frac{2}{n} trace(\Sigma^2) $$ where \(\lambda_i\) are the eigenvalues of \(\Sigma\).

This is a simplified version of the general CLT for linear spectral statistics. For i.i.d. components with finite fourth moments, the trace is asymptotically normal with this variance.

Asymptotic regime: The theory applies when both sample size \(n\) and dimension \(p\) can grow, typically requiring \(n >> p\) for good finite-sample performance. The approximation improves as the sample size increases.

Comparison with bootstrap: For small to moderate samples, bootstrap confidence intervals (see eigstatCI()) may provide better coverage. This asymptotic approach is faster but may be anticonservative (too narrow) when \(n\) is small relative to \(p\).

References

Bai, Z. D., & Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Annals of Probability, 32(1A), 553-605. tools:::Rd_expr_doi("10.1214/aop/1078415845")

Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Wiley-Interscience.

See Also

boxM(), plot.boxM(), logdetCI(), eigstatCI()

Examples

# Iris data example
data(iris)
iris.mod <- lm(as.matrix(iris[,1:4]) ~ iris$Species)
iris.boxm <- boxM(iris.mod)

# Get covariance matrices
cov <- c(iris.boxm$cov, list(pooled = iris.boxm$pooled))
n <- c(rep(50, 3), 150)

# Calculate trace CIs
CI <- traceCI(cov, n = n, conf = 0.95)
CI

# Compare with plot
plot(iris.boxm, which = "sum", gplabel = "Species",
     main = "Sum of eigenvalues (trace)")
arrows(CI$lower, 1:4, CI$upper, 1:4,
       lwd = 3, angle = 90, length = 0.1, code = 3, col = "red")

# Single covariance matrix
S <- cov(iris[,1:4])
traceCI(S, n = 150)

# Compare different confidence levels
traceCI(cov, n = n, conf = 0.90)
traceCI(cov, n = n, conf = 0.95)
traceCI(cov, n = n, conf = 0.99)