data matrix of pooled sample or Euclidean distances
sizes
vector of sample sizes
distance
logical: if TRUE, x is a distance matrix
ix
a permutation of the row indices of x
alpha
distance exponent in (0,2]
method
how to weight the statistics
Details
A vector containing the pairwise two-sample multivariate
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
|| || denotes Euclidean norm, a=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 (n_i n_j)(n_i+n_j)
is one-half of the harmonic mean of the sample sizes. The
discoB and discoF methods are related but
different ways of summarizing the pairwise differences between samples.
The disco methods apply the coefficient
(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.
Value
A object of class dist containing the lower triangle of the
e-distance matrix of cluster distances corresponding to the permutation
of indices ix is returned. The method attribute of the
distance object is assigned a value of type, index.
Szekely, G. J. and Rizzo, M. L. (2005) Hierarchical Clustering
via Joint Between-Within Distances: Extending Ward's Minimum
Variance Method, Journal of Classification 22(2) 151-183.
http://dx.doi.org/10.1007/s00357-005-0012-9
M. L. Rizzo and G. J. Szekely (2010).
DISCO Analysis: A Nonparametric Extension of
Analysis of Variance, Annals of Applied Statistics,
Vol. 4, No. 2, 1034-1055.
"http://dx.doi.org/10.1214/09-AOAS245"
Szekely, G. J. and Rizzo, M. L. (2004) Testing for Equal
Distributions in High Dimension, InterStat, November (5).
Szekely, G. J. (2000) Technical Report 03-05,
E-statistics: Energy of
Statistical Samples, Department of Mathematics and Statistics,
Bowling Green State University.
See Also
energy.hclusteqdist.etestksample.edisco
Examples
## compute cluster e-distances for 3 samples of iris data
data(iris)
edist(iris[,1:4], c(50,50,50))
## pairwise disco statistics
edist(iris[,1:4], c(50,50,50), method="discoF") #F ratios
## compute e-distances from vector of group labels
d <- dist(matrix(rnorm(100), nrow=50))
g <- cutree(energy.hclust(d), k=4)
edist(d, sizes=table(g), ix=rank(g, ties.method="first"))