Performs the Biswas and Ghosh (2014) two-sample test for high-dimensional data.
BG2(X1, X2, n.perm = 0, seed = 42)An object of class htest with the following components:
Observed value of the test statistic
Asymptotic or permutation 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
Number of permutations for permutation test (default: 0, asymptotic test is performed).
Random seed (default: 42)
| Target variable? | Numeric? | Categorical? | K-sample? |
| No | Yes | No | No |
The test is based on comparing the means of the distributions of the within-sample and between-sample distances of both samples. It is intended for the high dimension low sample size (HDLSS) setting and claimed to perform better in this setting than the tests of Friedman and Rafsky (1979), Schilling (1986) and Henze (1988) and the Cramér test of Baringhaus and Franz (2004).
The statistic is defined as $$T = ||\hat{\mu}_{D_F} - \hat{\mu}_{D_G}||^2_2, \text{ where}$$ $$\hat{\mu}_{D_F} = \left[\hat{\mu}_{FF} = \frac{2}{n_1(n_1 - 1)}\sum_{i=1}^{n_1}\sum_{j=i+1}^{n_1}||X_{1i} - X_{1j}||, \hat{\mu}_{FG} = \frac{1}{n_1 n_2}\sum_{i=1}^{n_1}\sum_{j=1}^{n_2}||X_{1i} - X_{2j}||\right], $$ $$\hat{\mu}_{D_G} = \left[\hat{\mu}_{FG} = \frac{1}{n_1 n_2}\sum_{i=1}^{n_1}\sum_{j=1}^{n_2}||X_{1i} - X_{2j}||, \hat{\mu}_{GG} = \frac{2}{n_2(n_2 - 1)}\sum_{i=1}^{n_2}\sum_{j=i+1}^{n_2}||X_{2i} - X_{2j}||\right]. $$
For testing, the scaled statistic $$T^* = \frac{N\hat{\lambda}(1 - \hat{\lambda})}{2\hat{\sigma}_0^2} T \text{ with}$$ $$\hat{\lambda} = \frac{n_1}{N},$$ $$\hat{\sigma}_0^2 = \frac{n_1S_1 + n_2S_2}{N}, \text{ where}$$ $$S_1 = \frac{1}{\binom{n_1}{3}} \sum_{1\le i < j < k \le n_1} ||X_{1i} - X_{1j}||\cdot ||X_{1i} - X_{1k}|| - \hat{\mu}_{FF}^2 \text{ and}$$ $$S_2 = \frac{1}{\binom{n_2}{3}} \sum_{1\le i < j < k \le n_2} ||X_{2i} - X_{2j}||\cdot ||X_{2i} - X_{2k}|| - \hat{\mu}_{GG}^2$$ is used as it is asymptotically \(\chi^2_1\)-distributed.
In both cases, low values indicate similarity of the datasets. Thus, the test rejects the null hypothesis of equal distributions for large values of the test statistic.
For n.perm > 0, a permutation test is conducted. Otherwise, an asymptotic test using the asymptotic distibution of \(T^*\) is performed.
Biswas, M., Ghosh, A.K. (2014). A nonparametric two-sample test applicable to high dimensional data. Journal of Multivariate Analysis, 123, 160-171, tools:::Rd_expr_doi("10.1016/j.jmva.2013.09.004").
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")
Energy, Cramer
# Draw some data
X1 <- matrix(rnorm(1000), ncol = 10)
X2 <- matrix(rnorm(1000, mean = 0.5), ncol = 10)
# Perform Biswas and Ghosh test
BG2(X1, X2)
Run the code above in your browser using DataLab