subcopem: Bivariate Empirical Subcopula

In subcopem2D: Bivariate Empirical Subcopula

Description

Calculation of bivariate empirical subcopula matrix, induced partitions, standardized bivariate sample, and dependence measures for a given bivariate sample.

Usage

subcopem(mat.xy, display = FALSE)

Arguments

mat.xy	2-column matrix with bivariate observations of a random vector (X,Y).
display	logical value indicating if graphs and dependence measures should be displayed.

Details

Each of the random variables X and Y may be of any kind (discrete, continuous, or mixed). NA values are not allowed.

Value

A list containing the following components:

depMon	monotone standardized supremum distance in [-1,1].
depMonNonSTD	monotone non-standardized supremum distance [min,value,max].
depSup	standardized supremum distance in [0,1].
depSupNonSTD	non-standardized supremum distance [min,value,max].
matrix	matrix with empirical subcopula values.
part1	vector with partition induced by first variable X.
part2	vector with partition induced by second variable Y.
sample.size	numeric value of sample size.
std.sample	2-column matrix with the standardized bivariate sample.
sample	2-column matrix with the original bivariate sample of (X,Y).

If display = TRUE then the values of depMon, depMonNonSTD, depSup, and depSupNonSTD will be displayed, and the following graphs will be generated: marginal histograms of X and Y, scatterplots of the original and the standardized bivariate sample, contour and image bivariate graphs of the empirical subcopula.

Note

If both X and Y are continuous random variables it is faster and better to use subcopemc.

Author(s)

Arturo Erdely https://sites.google.com/site/arturoerdely

References

Durante, F. and Sempi, C. (2016) Principles of Copula Theory. Taylor and Francis Group, Boca Raton.

Erdely, A. (2017) A subcopula based dependence measure. Kybernetika 53(2), 231-243. DOI: 10.14736/kyb-2017-2-0231

Nelsen, R.B. (2006) An Introduction to Copulas. Springer, New York.

See Also

subcopemc

Examples

## Example 1: Discrete-discrete Poisson positive dependence
n <- 1000  # sample size
X <- rpois(n, 5)  # Poisson(parameter = 5)
p <- 2  # another parameter
Y <- mapply(rpois, rep(1, n), 1 + p*X)  # creating dependence
XY <- cbind(X, Y)  # 2-column matrix with bivariate sample
cor(XY, method = "pearson")[1, 2]   # Pearson's correlation
cor(XY, method = "spearman")[1, 2]  # Spearman's correlation
cor(XY, method = "kendall")[1, 2]  # Kendall's correlation
SC <- subcopem(XY, display = TRUE)
str(SC)

## Example 2: Continuous-discrete non-monotone dependence
n <- 1000      # sample size
X <- rnorm(n)                  # Normal(0,1)
Y <- 2*(X > 1) - 1*(X > -1)    # Discrete({-1, 0, 1})
XY <- cbind(X, Y)  # 2-column matrix with bivariate sample
cor(XY, method = "pearson")[1, 2]   # Pearson's correlation
cor(XY, method = "spearman")[1, 2]  # Spearman's correlation
cor(XY, method = "kendall")[1, 2]  # Kendall's correlation
SC <- subcopem(XY, display = TRUE)
str(SC)

Example output

[1] 0.8099064
[1] 0.7924834
[1] 0.6495426
monotone dependence = [ -1 , 0.61634802 , 1 ]
non-std interval = [ -0.927072 , 0.600392 , 0.974112 ]
supremum dependence = [ 0 , 0.61634802 , 1 ]
non-std interval = [ 0 , 0.600392 , 0.974112 ]
List of 10
 $ depMon      : num 0.616
 $ depMonNonSTD: Named num [1:3] -0.927 0.6 0.974
  ..- attr(*, "names")= chr [1:3] "min" "value" "max"
 $ depSup      : num 0.616
 $ depSupNonSTD: Named num [1:3] 0 0.6 0.974
  ..- attr(*, "names")= chr [1:3] "min" "value" "max"
 $ matrix      : num [1:15, 1:32] 0 0 0 0 0 0 0 0 0 0 ...
 $ part1       : num [1:15] 0 0.005 0.04 0.141 0.284 0.444 0.617 0.771 0.867 0.936 ...
 $ part2       : num [1:32] 0 0.004 0.019 0.031 0.059 0.1 0.159 0.224 0.281 0.374 ...
 $ sample.size : int 1000
 $ std.sample  : num [1:1000, 1:2] 0.771 0.771 0.771 0.867 0.968 0.444 0.867 0.936 0.284 0.968 ...
 $ sample      : int [1:1000, 1:2] 6 6 6 7 9 4 7 8 3 9 ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : NULL
  .. ..$ : chr [1:2] "X" "Y"
[1] 0.2972084
[1] 0.1150596
[1] 0.04113666
monotone dependence = [ -1 , 0.06595935 , 1 ]
non-std interval = [ -0.867504 , 0.05722 , 0.867504 ]
supremum dependence = [ 0 , 0.59111658 , 1 ]
non-std interval = [ 0 , 0.512796 , 0.867504 ]
dev.new(): using pdf(file="Rplots1.pdf")
List of 10
 $ depMon      : num 0.066
 $ depMonNonSTD: Named num [1:3] -0.8675 0.0572 0.8675
  ..- attr(*, "names")= chr [1:3] "min" "value" "max"
 $ depSup      : num 0.591
 $ depSupNonSTD: Named num [1:3] 0 0.513 0.868
  ..- attr(*, "names")= chr [1:3] "min" "value" "max"
 $ matrix      : num [1:1001, 1:4] 0 0 0 0 0 0 0 0 0 0 ...
 $ part1       : num [1:1001] 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 ...
 $ part2       : num [1:4] 0 0.682 0.849 1
 $ sample.size : int 1000
 $ std.sample  : num [1:1000, 1:2] 0.558 0.726 0.228 0.95 0.564 0.855 0.356 0.397 0.305 0.486 ...
 $ sample      : num [1:1000, 1:2] 0.12 0.566 -0.739 1.655 0.126 ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : NULL
  .. ..$ : chr [1:2] "X" "Y"

subcopem2D documentation built on May 2, 2019, 2:46 p.m.