# Ball Statistics ### Introduction

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.

### Installation

#### CRAN version

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

``install.packages("Ball")``

#### Github version

To install the development version from GitHub, run:

``````library(devtools)
install_github("Mamba413/Ball", build_vignettes = TRUE)``````

Windows user will need to install Rtools first.

### Usage

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

#### bd.test

``````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:

``````    Nonparametric 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.

#### bcov.test

``````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.

#### bcorsis

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)
``#     3    1 1601   20  429``