The fundamental problems for data mining and statistical analysis are:

Whether distributions of two samples are distinct?

Whether two random variables are dependent?

**Ball** package provides solutions for these issues. Moreover, a variable screening (or feature screening) procedure is also implemented to tackle ultra high dimensional data. The core functions in **Ball** package are **bd.test**, **bcov.test**, and **bcorsis**.

These functions based on ball statistic have several advantages:

It’s applicable to univariate and multivariate data in Banach space.

There is no need for moment assumption, which means that outliers and heavy-tail data are no longer a problem.

They perform well in many setting without complex adjustments for parameters.

Particularly, for two-sample or K-sample problem, **bd.test** has been proved to cope well for imbalanced data, and **bcov.test** and **bcorsis** work well for detecting the relationship between complex responses and/or predictors, such as shape, compositional as well as censored data.

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

To install the development version from GitHub, run:

*Windows* user will need to install Rtools first.

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-Samples Ball Divergence Test
data: virginica and versicolor
number of observations = 100, group sizes: 50 50
replicates = 99
bd = 0.32912, 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:

```
Ball Covariance test of independence
data: sepal and petal
number of observations = 150
replicates = 99, Weighted Ball Covariance = FALSE
bcov = 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:
set.seed(1)
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 find any bugs, or if you experience any crashes, please report to us. If you have any questions just ask, we won’t bite.

GPL-3

- Pan, Wenliang; Tian, Yuan; Wang, Xueqin; Zhang, Heping. Ball Divergence: Nonparametric two sample test. Ann. Statist. 46 (2018), no. 3, 1109–1137. doi:10.1214/17-AOS1579. https://projecteuclid.org/euclid.aos/1525313077
- Wenliang Pan, Xueqin Wang, Weinan Xiao & Hongtu Zhu (2018) A Generic Sure Independence Screening Procedure, Journal of the American Statistical Association, DOI: 10.1080/01621459.2018.1462709
- Wenliang Pan, Xueqin Wang, Heping Zhang, Hongtu Zhu & Jin Zhu (2019) Ball Covariance: A Generic Measure of Dependence in Banach Space, Journal of the American Statistical Association, DOI: 10.1080/01621459.2018.1543600
- Jin, Z., Wenliang P., Wei Z., and Xueqin W. (2018). Ball: An R package for detecting distribution difference and association in metric spaces. arXiv preprint arXiv:1811.03750. URL http://arxiv.org/abs/1811.03750.