A vector containing the pairwise two-sample multivariate
\(\mathcal{E}\)-statistics for comparing clusters or samples is returned.
The e-distance between clusters is computed from the original pooled data,
stacked in matrix `x`

where each row is a multivariate observation, or
from the distance matrix `x`

of the original data, or distance object
returned by `dist`

. The first `sizes[1]`

rows of the original data
matrix are the first sample, the next `sizes[2]`

rows are the second
sample, etc. The permutation vector `ix`

may be used to obtain
e-distances corresponding to a clustering solution at a given level in
the hierarchy.

The default method `cluster`

summarizes the e-distances between
clusters in a table.
The e-distance between two clusters \(C_i, C_j\)
of size \(n_i, n_j\)
proposed by Szekely and Rizzo (2005)
is the e-distance \(e(C_i,C_j)\), defined by
$$e(C_i,C_j)=\frac{n_i n_j}{n_i+n_j}[2M_{ij}-M_{ii}-M_{jj}],
$$
where
$$M_{ij}=\frac{1}{n_i n_j}\sum_{p=1}^{n_i} \sum_{q=1}^{n_j}
\|X_{ip}-X_{jq}\|^\alpha,$$
\(\|\cdot\|\) denotes Euclidean norm, \(\alpha=\)
`alpha`

, and \(X_{ip}\) denotes the p-th observation in the i-th cluster. The
exponent `alpha`

should be in the interval (0,2].

The coefficient \(\frac{n_i n_j}{n_i+n_j}\)
is one-half of the harmonic mean of the sample sizes. The
`discoB`

method is related but with
different ways of summarizing the pairwise differences between samples.
The `disco`

methods apply the coefficient
\(\frac{n_i n_j}{2N}\) where N is the total number
of observations. This weights each (i,j) statistic by sample size
relative to N. See the `disco`

topic for more details.