Generalized Variance by Subgroup

Compute the generalized variance, defined as the determinant of the unbiased sample covariance matrix, for each multivariate subgroup.

Usage

gv_stat(x, size = NULL)

Arguments

x

Multivariate subgroup data in one of three forms:

a non-empty list whose elements are $n \times p$ numeric matrices;
a numeric three-dimensional array with dimensions subgroup by observation by variable; or
a numeric matrix formed by stacking equal-sized subgroups in consecutive blocks of rows.

Every observation must be finite.

size

Positive integer subgroup size. It is required when x is a stacked matrix and ignored for list or array input. The number of matrix rows must be an exact multiple of size.

Value

A numeric vector of length equal to the number of subgroups. Element $i$ is $|S_i|$. A determinant that is slightly negative only because of floating-point roundoff is returned as zero.

Details

For subgroup $i$ with $n$ observations on $p$ variables, gv_stat() calculates $$|S_i|, \qquad S_i = \frac{1}{n-1} \sum_{j=1}^{n}(x_{ij}-\bar{x}_i)(x_{ij}-\bar{x}_i)^T.$$ All subgroups must have the same $n$ and $p$, with $p >= 2$ and $n > p$. The latter condition permits a nonsingular sample covariance matrix under nondegenerate data, although an observed subgroup may still have determinant zero because of exact linear dependence.

Errors

An error is raised for unsupported input types, empty or unequal-sized subgroups, non-finite observations, fewer than two variables, $n <= p$, an invalid matrix size, or a covariance determinant that is genuinely negative.

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))

gv_stat(list(g1, g2))
#> [1]  1.00000 20.38889
gv_stat(rbind(g1, g2), size = 4)
#> [1]  1.00000 20.38889

x <- array(c(g1, g2), dim = c(2, 4, 2))
gv_stat(x)
#> [1] 1.500000 4.944444