Dispersion of Ensemble Members
measureDispersion.RdComputes, sample by sample and mode by mode, the sample variance of the members of an ensemble around their mean, and that sample variance divided by the number of members. The latter estimates the variance of the mean under the assumptions stated below.
Details
With \(M\) realizations \(C^{(m)}\) and their mean \(\bar C\), the sample variance is $$s^2 = \frac{1}{M - 1} \sum_{m=1}^{M} \left(C^{(m)} - \bar C\right)^2$$ and the returned estimate for the variance of the mean is \(s^2 / M\). The first describes dispersion between members. Interpreting the second as Monte Carlo precision assumes comparable components and independent realizations with a common mean and a common finite covariance. Opposite perturbations in a complementary pair are dependent; the two signs do not provide independent realizations for this interpretation.
The function does not establish these statistical assumptions or physical
correspondence between equal mode indices. Dispersion under perturbations
of a fixed record does not quantify all uncertainty in the physical signal,
and increasing the number of members does not remove decomposition bias.
Attributes of realizations other than dim do not reach the
outputs.
Value
A list with two matrices, samples by modes:
- variance
The sample variance \(s^2\), in squared units of the signal.
- meanVariance
The estimate \(s^2 / M\), subject to the assumptions in Details.
Examples
# Two realizations of four samples: the first has two modes, the second
# one, so its second mode is filled with zeros.
Realizations <- array(c(2, 4, 6, 8, 1, 1, 1, 1,
4, 6, 8, 10, 0, 0, 0, 0), dim = c(4, 2, 2))
measureDispersion(Realizations)
#> $variance
#> [,1] [,2]
#> [1,] 2 0.5
#> [2,] 2 0.5
#> [3,] 2 0.5
#> [4,] 2 0.5
#>
#> $meanVariance
#> [,1] [,2]
#> [1,] 1 0.25
#> [2,] 1 0.25
#> [3,] 1 0.25
#> [4,] 1 0.25
#>