SH: Schilling-Henze Nearest Neighbor Test

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

Schilling-Henze Nearest Neighbor Test

Description

Performs the Schilling-Henze two-sample test for multivariate data (Schilling, 1986; Henze, 1988).

Usage

SH(X1, X2, K = 5, graph.fun = knn.bf, dist.fun = stats::dist, n.perm = 0, 
    dist.args = NULL, seed = NULL)

Arguments

X1	First dataset as matrix or data.frame
X2	Second dataset as matrix or data.frame
K	Number of nearest neighbors to consider (default: 5)
graph.fun	Function for calculating a similarity graph using the distance matrix on the pooled sample (default: knn.bf which searches for the K nearest neighbors by ranking all pairwise distances, alternative: knn which is a wrapper for extracting the edge matrix from the result of kNN in dbscan, knn.fast which is a wrapper for the approximative KNN implementation get.knn in FNN, or any other function that calculates the KNN edge matrix from a distance matrix and the number of nearest neighbors K).
dist.fun	Function for calculating a distance matrix on the pooled dataset (default: stats::dist, Euclidean distance).
n.perm	Number of permutations for permutation test (default: 0, asymptotic test is performed).
dist.args	Named list of further arguments passed to dist.fun.
seed	Random seed (default: NULL). A random seed will only be set if one is provided.

Details

The test statistic is the proportion of edges connecting points from the same dataset in a K-nearest neighbor graph calculated on the pooled sample (standardized with expectation and SD under the null).

Low values of the test statistic indicate similarity of the datasets. Thus, the null hypothesis of equal distributions is rejected for high values.

For n.perm = 0, an asymptotic test using the asymptotic normal approximation of the conditional null distribution is performed. For n.perm > 0, a permutation test 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
estimate	The number of within-sample edges
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

Note

The default for K is chosen based on simulation results of Stolte et al. (2026). Note that still there is little guidance on the choice of K. Typical values for K chosen in the literature are 1 and 5.

Because this method cannot handle missing data, any missing values are removed automatically and a warning is issued.

References

Schilling, M. F. (1986). Multivariate Two-Sample Tests Based on Nearest Neighbors. Journal of the American Statistical Association, 81(395), 799-806. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/2289012")}

Henze, N. (1988). A Multivariate Two-Sample Test Based on the Number of Nearest Neighbor Type Coincidences. The Annals of Statistics, 16(2), 772-783.

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")}

Stolte, M., Rahnenführer, J., Bommert, A. (2026). An Empirical Comparison of Methods for Quantifying the Similarity of Numeric Datasets. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.48550/arXiv.2604.12327")}

See Also

knn, BQS, FR, CF, CCS, ZC for other graph-based tests, FR_cat, CF_cat, CCS_cat, and ZC_cat for versions of the test for categorical data

Examples

set.seed(1234)
# Draw some data
X1 <- matrix(rnorm(1000), ncol = 10)
X2 <- matrix(rnorm(1000, mean = 0.5), ncol = 10)
# Perform Schilling-Henze test
SH(X1, X2)

DataSimilarity documentation built on May 15, 2026, 9:07 a.m.