Trace Statistic for Multivariate Variability

Compute the auxiliary trace statistic $$T = (n - 1) \operatorname{tr}(\Sigma_0^{-1} S),$$ where $S$ is the unbiased subgroup covariance matrix and $\Sigma_0$ is the in-control covariance matrix.

Usage

trv_stat(x, size = NULL, Sigma0)

Arguments

x: Multivariate subgroup data in any format accepted by gv_stat: a list of numeric matrices, a subgroup-by-observation-by-variable array, or a stacked numeric matrix.
size: Positive integer subgroup size when x is a stacked matrix. It is ignored for list and array input.
Sigma0: Finite symmetric positive-definite in-control covariance matrix.

Value

A numeric vector with one trace statistic per subgroup.

Details

The statistic complements the generalized variance $|S|$. Two covariance matrices can have the same determinant but different trace after standardization by $\Sigma_0$; trv_stat() is designed to make that structural change visible.

Under independent sampling from a multivariate normal process with covariance $\Sigma_0$, $$(n - 1)S \sim W_p(n - 1, \Sigma_0)$$ and therefore $$(n - 1)\operatorname{tr}(\Sigma_0^{-1}S) \sim \chi^2_{p(n - 1)}.$$ The implementation evaluates the trace as (n - 1) * sum(Sigma0_inverse * S), avoiding an explicit matrix square root while preserving the same value.

Errors

Errors are raised for invalid subgroup data, non-finite observations, subgroup sizes smaller than two, a non-square or non-symmetric Sigma0, a dimension mismatch, or a covariance matrix that is not positive definite.

References

Barbosa, E. P., Gneri, M. A., and Meneguetti, A. Improving Shewhart-type Generalized Variance Control Charts for Multivariate Process Variability Monitoring using Cornish-Fisher Quantile Correction, Meijer-G Function and Other Tools. Research report.

Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis, 2nd ed. Wiley.

Examples

g1 <- cbind(c(0, 1, 2, 3), c(0, 2, 1, 3))
g2 <- cbind(c(1, 2, 4, 7), c(3, 1, 5, 2))
trv_stat(list(g1, g2), Sigma0 = diag(2))
#> [1] 10.00 29.75