Ball Covariance Test

Ball Covariance test of independence. Ball covariance are generic dependence measures in Banach spaces.

bcov.test(x, ...)

# S3 method for default
bcov.test(
  x,
  y = NULL,
  num.permutations = 99,
  method = c("permutation", "limit"),
  distance = FALSE,
  weight = FALSE,
  seed = 1,
  num.threads = 0,
  ...
)

# S3 method for formula
bcov.test(formula, data, subset, na.action, ...)

Arguments

x: a numeric vector, matrix, data.frame, or a list containing at least two numeric vectors, matrices, or data.frames.
...: further arguments to be passed to or from methods.
y: a numeric vector, matrix, or data.frame.
num.permutations: the number of permutation replications. When num.permutations = 0, the function just returns the Ball Covariance statistic. Default: num.permutations = 99.
method: if method = "permutation", a permutation procedure is carried out to compute the \(p\)-value; if method = "limit", an approximate null distribution is used when weight = "constant". Any unambiguous substring can be given. Default method = "permutation".
distance: if distance = TRUE, the elements of x and y are considered as distance matrices.
weight: a logical or character string used to choose the weight form of Ball Covariance statistic.. If input is a character string, it must be one of "constant", "probability", or "chisquare". Any unambiguous substring can be given. If input is a logical value, it is equivalent to weight = "probability" if weight = TRUE while equivalent to weight = "constant" if weight = FALSE. Default: weight = FALSE.
seed: the random seed. Default seed = 1.
num.threads: number of threads. If num.threads = 0, then all of available cores will be used. Default num.threads = 0.
formula: a formula of the form ~ u + v, where each of u and v are numeric variables giving the data values for one sample. The samples must be of the same length.
data: an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset: an optional vector specifying a subset of observations to be used.
na.action: a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Value

If num.permutations > 0, bcov.test returns a htest class object containing the following components:

statistic: Ball Covariance statistic.
p.value: the p-value for the test.
replicates: permutation replications of the test statistic.
size: sample size.
complete.info: a list mainly containing two vectors, the first vector is the Ball Covariance statistics with different weights, the second is the \(p\)-values of weighted Ball Covariance tests.
alternative: a character string describing the alternative hypothesis.
method: a character string indicating what type of test was performed.
data.name: description of data.

If num.permutations = 0, bcov.test returns a statistic value.

Details

bcov.test is non-parametric tests of independence in Banach spaces. It can detect the dependence between two random objects (variables) and the mutual dependence among at least three random objects (variables).

If two samples are pass to arguments x and y, the sample sizes (i.e. number of rows or length of the vector) of the two variables must agree. If a list object is passed to x, this list must contain at least two numeric vectors, matrices, or data.frames, and each element of this list must with the same sample size. Moreover, data pass to x or y must not contain missing or infinite values. If distance = TRUE, x is considered as a distance matrix or a list containing distance matrices, and y is considered as a distance matrix; otherwise, these arguments are treated as data.

bcov.test utilizes the Ball Covariance statistics (see bcov) to measure dependence and derives a \(p\)-value via replicating the random permutation num.permutations times.

See Pan et al 2018 for theoretical properties of the test, including statistical consistency.

Note

Actually, bcov.test simultaneously computing Ball Covariance statistics with "constant", "probability", and "chisquare" weights. Users can get other Ball Covariance statistics with different weight and their corresponding \(p\)-values in the complete.info element of output. We give a quick example below to illustrate.

References

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 Zhu, Wenliang Pan, Wei Zheng, and Xueqin Wang (2021). Ball: An R Package for Detecting Distribution Difference and Association in Metric Spaces, Journal of Statistical Software, Vol.97(6), doi: 10.18637/jss.v097.i06.

Author

Wenliang Pan, Xueqin Wang, Heping Zhang, Hongtu Zhu, Jin Zhu

Examples

set.seed(1)

################# Quick Start #################
noise <- runif(50, min = -0.3, max = 0.3)
x <- runif(50, 0, 4*pi)
y <- cos(x) + noise
# plot(x, y)
res <- bcov.test(x, y)
res
#> 
#> 	Ball Covariance test of independence (Permutation)
#> 
#> data:  x and y
#> number of observations = 50
#> replicates = 99, weight: constant
#> bcov.constant = 0.0021965, p-value = 0.01
#> alternative hypothesis: random variables are dependent
#> 
## get all Ball Covariance statistics:
res[["complete.info"]][["statistic"]]
#>    bcov.constant bcov.probability   bcov.chisquare 
#>      0.002196459      0.052685015      0.102469543 
## get all test result:
res[["complete.info"]][["p.value"]]
#>    bcov.constant.pvalue bcov.probability.pvalue   bcov.chisquare.pvalue 
#>                    0.01                    0.01                    0.01 

################# Quick Start #################
x <- matrix(runif(50 * 2, -pi, pi), nrow = 50, ncol = 2)
noise <- runif(50, min = -0.1, max = 0.1)
y <- sin(x[,1] + x[,2]) + noise
bcov.test(x = x, y = y, weight = "prob")
#> 
#> 	Ball Covariance test of independence (Permutation)
#> 
#> data:  x and y
#> number of observations = 50
#> replicates = 99, weight: probability
#> bcov.probability = 0.037912, p-value = 0.01
#> alternative hypothesis: random variables are dependent
#> 

################# Ball Covariance Test for Non-Hilbert Data #################
# load data:
data("ArcticLake")
# Distance matrix between y:
Dy <- nhdist(ArcticLake[["x"]], method = "compositional")
# Distance matrix between x:
Dx <- dist(ArcticLake[["depth"]])
# hypothesis test with BCov:
bcov.test(x = Dx, y = Dy, distance = TRUE)
#> 
#> 	Ball Covariance test of independence (Permutation)
#> 
#> data:  Dx and Dy
#> number of observations = 39
#> replicates = 99, weight: constant
#> bcov.constant = 0.0083848, p-value = 0.01
#> alternative hypothesis: random variables are dependent
#> 

################  Weighted Ball Covariance Test  #################
data("ArcticLake")
Dy <- nhdist(ArcticLake[["x"]], method = "compositional")
Dx <- dist(ArcticLake[["depth"]])
# hypothesis test with weighted BCov:
bcov.test(x = Dx, y = Dy, distance = TRUE, weight = "prob")
#> 
#> 	Ball Covariance test of independence (Permutation)
#> 
#> data:  Dx and Dy
#> number of observations = 39
#> replicates = 99, weight: probability
#> bcov.probability = 0.088597, p-value = 0.01
#> alternative hypothesis: random variables are dependent
#> 

################# Mutual Independence Test #################
x <- rnorm(50)
y <- (x > 0) * x + rnorm(50)
z <- (x <= 0) * x + rnorm(50)
data_list <- list(x, y, z)
bcov.test(data_list)
#> 
#> 	Ball Covariance test of mutual independence (Permutation)
#> 
#> data:  data_list
#> number of observations = 50
#> replicates = 99, weight: constant
#> bcov.constant = 0.0016842, p-value = 0.03
#> alternative hypothesis: random variables are dependent
#> 
data_list <- lapply(data_list, function(x) {
  as.matrix(dist(x))
})
bcov.test(data_list, distance = TRUE)
#> 
#> 	Ball Covariance test of mutual independence (Permutation)
#> 
#> data:  data_list
#> number of observations = 50
#> replicates = 99, weight: constant
#> bcov.constant = 0.0016842, p-value = 0.03
#> alternative hypothesis: random variables are dependent
#> 
bcov.test(data_list, distance = FALSE, weight = "chi")
#> 
#> 	Ball Covariance test of mutual independence (Permutation)
#> 
#> data:  data_list
#> number of observations = 50
#> replicates = 99, weight: chisquare
#> bcov.chisquare = 1.7114, p-value = 0.03
#> alternative hypothesis: random variables are dependent
#> 

################# Mutual Independence Test for Meteorology data #################
data("meteorology")
bcov.test(meteorology)
#> 
#> 	Ball Covariance test of mutual independence (Permutation)
#> 
#> data:  meteorology
#> number of observations = 46
#> replicates = 99, weight: constant
#> bcov.constant = 0.0064547, p-value = 0.01
#> alternative hypothesis: random variables are dependent
#> 

################  Testing via approximate limit distribution  #################
if (FALSE) {
set.seed(1)
n <- 2000
x <- rnorm(n)
y <- rnorm(n)
bcov.test(x, y, method = "limit")
bcov.test(x, y)
}

################  Formula interface  ################
## independence test:
bcov.test(~ CONT + INTG, data = USJudgeRatings)
#> 
#> 	Ball Covariance test of independence (Permutation)
#> 
#> data:  CONT and INTG
#> number of observations = 43
#> replicates = 99, weight: constant
#> bcov.constant = 0.00079459, p-value = 0.39
#> alternative hypothesis: random variables are dependent
#> 
## independence test with chisquare weight:
bcov.test(~ CONT + INTG, data = USJudgeRatings, weight = "chi")
#> 
#> 	Ball Covariance test of independence (Permutation)
#> 
#> data:  CONT and INTG
#> number of observations = 43
#> replicates = 99, weight: chisquare
#> bcov.chisquare = 0.038503, p-value = 0.3
#> alternative hypothesis: random variables are dependent
#> 
## mutual independence test:
bcov.test(~ CONT + INTG + DMNR, data = USJudgeRatings)
#> 
#> 	Ball Covariance test of mutual independence (Permutation)
#> 
#> data:  CONT and INTG and DMNR
#> number of observations = 43
#> replicates = 99, weight: constant
#> bcov.constant = 0.0078353, p-value = 0.01
#> alternative hypothesis: random variables are dependent
#>