Performs the Energy statistic multi-sample test (Székely and Rizzo, 2004). The implementation here uses the eqdist.etest implementation from the energy package.
Energy(X1, X2, ..., n.perm = 0, seed = 42)An object of class htest with the following components:
The function call
Observed value of the test statistic
Bootstrap p value
The alternative hypothesis
Description of the test
The dataset names
First dataset as matrix or data.frame
Second dataset as matrix or data.frame
Further datasets as matrices or data.frames
Number of permutations for Bootstrap test (default: 0, no Bootstrap test performed)
Random seed (default: 42)
| Target variable? | Numeric? | Categorical? | K-sample? |
| No | Yes | No | Yes |
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.
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. tools:::Rd_expr_doi("10.1214/24-SS149")
Cramer, DISCOB, DISCOF
# 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)
}
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