BG2: Biswas and Ghosh (2014) Two-Sample Test

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

Biswas and Ghosh (2014) Two-Sample Test

Description

Performs the Biswas and Ghosh (2014) two-sample test for high-dimensional data.

Usage

BG2(X1, X2, n.perm = 0, seed = 42)

Arguments

X1	First dataset as matrix or data.frame
X2	Second dataset as matrix or data.frame
n.perm	Number of permutations for permutation test (default: 0, asymptotic test is performed).
seed	Random seed (default: 42)

Details

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.

Value

An object of class htest with the following components:

statistic	Observed value of the test statistic
p.value	Asymptotic or permutation 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	No

References

Biswas, M., Ghosh, A.K. (2014). A nonparametric two-sample test applicable to high dimensional data. Journal of Multivariate Analysis, 123, 160-171, \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/24-SS149")}

See Also

Energy, Cramer

Examples

# 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)

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