Energy: Energy Statistic and Test

In DataSimilarity: Quantifying Similarity of Datasets and Multivariate Two- And k-Sample Testing

Energy Statistic and Test

Description

Performs the Energy statistic multi-sample test (Székely and Rizzo, 2004). The implementation here uses the eqdist.etest implementation from the energy package.

Usage

Energy(X1, X2, ..., n.perm = 0, seed = 42)

Arguments

X1	First dataset as matrix or data.frame
X2	Second dataset as matrix or data.frame
...	Further datasets as matrices or data.frames
n.perm	Number of permutations for Bootstrap test (default: 0, no Bootstrap test performed)
seed	Random seed (default: 42)

Details

The Energy statistic (Székely and Rizzo, 2004) for two datasets X_1 and X_2 is defined as

T_{n_1, n_2} = \frac{n_1 n_2}{n_1+n_2}\left(\frac{1}{n_1 n_2}\sum_{i=1}^{n_1}\sum_{j=1}^{n_2} ||X_{1i} - X_{2j}|| - \frac{1}{2n_1^2}\sum_{i,j=1}^{n_1} ||X_{1i} - X_{1j}|| - \frac{1}{2n_2^2}\sum_{i,j=1}^{n_2} ||X_{2i} - X_{2j}||\right).

This is equal to the Cramér test statistitic (Baringhaus and Franz, 2004). The multi-sample version is defined as the sum of the Energy statistics for all pairs of samples.

The population Energy statistic for two distributions is equal to zero if and only if the two distributions coincide. Therefore, small values of the empirical statistic indicate similarity between datasets and the permutation test rejects the null hypothesis of equal distributions for large values.

This implementation is a wrapper function around the function eqdist.etest that modifies the in- and output of that function to match the other functions provided in this package. For more details see the eqdist.etest.

Value

An object of class htest with the following components:

call	The function call
statistic	Observed value of the test statistic
p.value	Bootstrap p value
alternative	The alternative hypothesis
method	Description of the test
data.name	The dataset names

Applicability

Target variable?	Numeric?	Categorical?	K-sample?
No	Yes	No	Yes

Note

The test based on the Energy statistic (Székely and Rizzo, 2004) is equivalent to the Cramér test (Baringhaus and Franz, 2004).

References

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.

Rizzo, M., Szekely, G. (2022). energy: E-Statistics: Multivariate Inference via the Energy of Data. R package version 1.7-11, https://CRAN.R-project.org/package=energy.

Stolte, M., Kappenberg, F., Rahnenführer, J., Bommert, A. (2024). Methods for quantifying dataset similarity: a review, taxonomy and comparison. Statist. Surv. 18, 163 - 298. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/24-SS149")}

See Also

Cramer, DISCOB, DISCOF

Examples

# Draw some data
X1 <- matrix(rnorm(1000), ncol = 10)
X2 <- matrix(rnorm(1000, mean = 0.5), ncol = 10)
# Perform Energy test
if(requireNamespace("energy", quietly = TRUE)) {
  Energy(X1, X2, n.perm = 100)
}

DataSimilarity documentation built on April 3, 2025, 9:39 p.m.