BG2 {DataSimilarity} | R Documentation |
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 = NULL)
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: NULL). A random seed will only be set if one is provided. |
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, 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. doi:10.1214/24-SS149
See Also
Examples
set.seed(1234)
# 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)