Ball Statistics in R

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The fundamental problems for data mining, statistical analysis, and machine learning are: - whether several distributions are different? - whether random variables are dependent? - how to pick out useful variables/features from a high-dimensional data?

These issues can be tackled by using bd.test, bcov.test, and bcorsis functions in the Ball package, respectively. They enjoy following admirable advantages: - available for most of datasets (e.g., traditional tabular data, brain shape, functional connectome, wind direction and so on) - insensitive to outliers, distribution-free and model-free; - theoretically guaranteed and computationally efficient.


CRAN version

To install the Ball R package from CRAN, just run:


Github version

To install the development version from GitHub, run:

install_github("Mamba413/Ball/R-package", build_vignettes = TRUE)

Windows user will need to install Rtools first.

Overview: Ball package

Three most importance functions in Ball:

bd.test bcov.test bcorsis
Feature Hypothesis test Hypothesis test Feature screening
Type Test of equal distributions Test of (joint) independence SIS and ISIS
Optional weight :heavy_check_mark: :heavy_check_mark: :heavy_check_mark:
Parallel programming :heavy_check_mark: :heavy_check_mark: :heavy_check_mark:
p-value :heavy_check_mark: :heavy_check_mark: :x:
Limit distribution Two-sample test only Independence test only :x:
Censored data :x: :x: :heavy_check_mark:
Interaction screening :x: :x: :heavy_check_mark:
GWAS optimization :x: :x: :heavy_check_mark:

Quick examples

Take iris dataset as an example to illustrate how to use bd.test and bcov.test to deal with the fundamental problems mentioned above.


virginica <- iris[iris$Species == "virginica", "Sepal.Length"]
versicolor <- iris[iris$Species == "versicolor", "Sepal.Length"]
bd.test(virginica, versicolor)

In this example, bd.test examines the assumption that Sepal.Length distributions of versicolor and virginica are equal.

If the assumption invalid, the p-value of the bd.test will be under 0.05.

In this example, the result is:

    2-sample Ball Divergence Test (Permutation)

data:  virginica and versicolor 
number of observations = 100, group sizes: 50 50
replicates = 99, weight: constant
bd.constant = 0.11171, p-value = 0.01
alternative hypothesis: distributions of samples are distinct

The R output shows that p-value is under 0.05. Consequently, we can conclude that the Sepal.Length distribution of versicolor and virginica are distinct.


sepal <- iris[, c("Sepal.Width", "Sepal.Length")]
petal <- iris[, c("Petal.Width", "Petal.Length")]
bcov.test(sepal, petal)

In this example, bcov.test investigates whether width or length of petal is associated with width and length of sepal. If the dependency really exists, the p-value of the bcov.test will be under 0.05. In this example, the result is show to be:

    Ball Covariance test of independence (Permutation)

data:  sepal and petal
number of observations = 150
replicates = 99, weight: constant
bcov.constant = 0.0081472, p-value = 0.01
alternative hypothesis: random variables are dependent

Therefore, the relationship between width and length of sepal and petal exists.


We generate a dataset and demonstrate the usage of bcorsis function as follow.

## simulate a ultra high dimensional dataset:
n <- 150
p <- 3000
x <- matrix(rnorm(n * p), nrow = n)
error <- rnorm(n)
y <- 3 * x[, 1] + 5 * (x[, 3])^2 + error

## BCor-SIS procedure:
res <- bcorsis(y = y, x = x)
head(res[["ix"]], n = 5)

In this example, the result is:

# [1]    3    1 1601   20  429

The bcorsis result shows that the first and the third variable are the two most important variables in 3000 explanatory variables which is consistent to the simulation settings.


If you use Ball or reference our vignettes in a presentation or publication, we would appreciate citations of our package. > Zhu J, Pan W, Zheng W, Wang X (2021). “Ball: An R Package for Detecting Distribution Difference and Association in Metric Spaces.” Journal of Statistical Software, 97(6), 1–31. doi: 10.18637/jss.v097.i06.

Here is the corresponding Bibtex entry

  title = {{Ball}: An {R} Package for Detecting Distribution Difference and Association in Metric Spaces},
  author = {Jin Zhu and Wenliang Pan and Wei Zheng and Xueqin Wang},
  journal = {Journal of Statistical Software},
  year = {2021},
  volume = {97},
  number = {6},
  pages = {1--31},
  doi = {10.18637/jss.v097.i06},


Bug report

If you find any bugs, or if you experience any crashes, please report to us. If you have any questions just ask, we won’t bite. Open an issue or send an email to Jin Zhu at