Package 'subcopem2D' reference manual

Title:	Bivariate Empirical Subcopula
Description:	Calculate empirical subcopula and dependence measures from a given bivariate sample, and Bernstein copula approximations.
Authors:	Arturo Erdely
Maintainer:	Arturo Erdely <[email protected]>
License:	GPL-3
Version:	1.3
Built:	2024-11-06 06:29:43 UTC
Source:	CRAN

Bernstein Copula Approximation

Description

Bernstein copula approximation from the empirical subcopula of given bivariate data.

Usage

Bcopula(mat.xy, m, both.cont = FALSE, tolimit = 1e-05)
Bcopula(mat.xy, m, both.cont = FALSE, tolimit = 1e-05)

Arguments

`mat.xy`	2-column matrix with bivariate observations of a random vector $(X,Y)$ .
`m`	integer value of approximation order, where $m = 2,...,n$ with $n$ equal to sample size. A recommended value for $m$ would be the minimum between $\sqrt{n}$ and $50$ .
`both.cont`	logical value, if TRUE then (X,Y) are considered (both) as continuos random variables, and jittering will be applied to repeated values (if any).
`tolimit`	tolerance limit in numerical approximation of the inverse of the first partial derivatives of the estimated Bernstein copula.

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:

`copula`	bivariate Bernstein Copula function (BC) of order $m$
`du`	bivariate function $\partial BC(u,v)/\partial u$
`dv`	bivariate function $\partial BC(u,v)/\partial v$
`du.inv`	inverse of `du` with respect to v, given u and alpha (numerical approx)
`dv.inv`	inverse of `dv` with respect to u, given v and alpha (numerical approx)
`density`	bivariate Bernstein copula density function of order $m$
`bilinearCopula`	bivariate function of bilinear approximation of copula
`bilinearSubcopula`	$(m+1)\times (m+1)$ matrix with empirical subcopula values
`sample.size`	sample size of bivariate observations
`order`	approximation order $m$ used
`both.cont`	logical value, TRUE if both variables considered as continuous
`tolimit`	tolerance limit in numerical approximation of `du.inv` and `dv.inv`
`subcopemObject`	list object with the output from `subcopem` if `both.cont = FALSE` or from `subcopemc` if `both.cont = TRUE`

Acknowledgement

Development of this code was partially supported by Programa UNAM DGAPA PAPIIT through project IN115817.

Note

If both $X$ and $Y$ are continuous random variables it is faster and better to set both.cont = TRUE.

Author(s)

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

References

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.

Sancetta, A., Satchell, S. (2004) The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20, 535-562. DOI: 10.1017/S026646660420305X

Examples

## (X,Y) continuous random variables with copula FGM(param = 1)

# Theoretical formulas
FGMcopula <- function(u, v) u*v*(1 + (1 - u)*(1 - v))
dFGM.du <- function(u, v) (2*u - 1)*(v^2) + 2*v*(1 - u)
dFGM.dv <- function(u, v) (2*v - 1)*(u^2) + 2*u*(1 - v)
A1 <- function(u) 2*(1 - u)
A2 <- function(u, z) sqrt(A1(u)^2 - 4*(A1(u) - 1)*z)
dFGM.du.inv <- function(u, z) 2*z/(A1(u) + A2(u, z))
FGMdensity <- function(u, v) 2*(1 - u - v + 2*u*v)

# Simulating FGM observations
n <- 3000
U <- runif(n)
Z <- runif(n)
V <- mapply(dFGM.du.inv, U, Z)

# Applying Bcopula to FGM simulated values
B <- Bcopula(cbind(U, V), 50, TRUE)
str(B)

# Comparing theoretical values versus Bernstein and Bilinear approximations
u <- 0.70; v <- 0.55
FGMcopula(u, v); B[["copula"]](u, v); B[["bilinearCopula"]](u, v)
dFGM.du(u, v); B[["du"]](u, v)
dFGM.dv(u, v); B[["dv"]](u, v)
dFGM.du.inv(u, 0.8); B[["du.inv"]](u, 0.8)
FGMdensity(u, v); B[["density"]](u, v)
## (X,Y) continuous random variables with copula FGM(param = 1)

# Theoretical formulas
FGMcopula <- function(u, v) u*v*(1 + (1 - u)*(1 - v))
dFGM.du <- function(u, v) (2*u - 1)*(v^2) + 2*v*(1 - u)
dFGM.dv <- function(u, v) (2*v - 1)*(u^2) + 2*u*(1 - v)
A1 <- function(u) 2*(1 - u)
A2 <- function(u, z) sqrt(A1(u)^2 - 4*(A1(u) - 1)*z)
dFGM.du.inv <- function(u, z) 2*z/(A1(u) + A2(u, z))
FGMdensity <- function(u, v) 2*(1 - u - v + 2*u*v)

# Simulating FGM observations
n <- 3000
U <- runif(n)
Z <- runif(n)
V <- mapply(dFGM.du.inv, U, Z)

# Applying Bcopula to FGM simulated values
B <- Bcopula(cbind(U, V), 50, TRUE)
str(B)

# Comparing theoretical values versus Bernstein and Bilinear approximations
u <- 0.70; v <- 0.55
FGMcopula(u, v); B[["copula"]](u, v); B[["bilinearCopula"]](u, v)
dFGM.du(u, v); B[["du"]](u, v)
dFGM.dv(u, v); B[["dv"]](u, v)
dFGM.du.inv(u, 0.8); B[["du.inv"]](u, 0.8)
FGMdensity(u, v); B[["density"]](u, v)

Dependence Measures

Description

Calculation of pairwise monotone and supremum dependence, monotone/supremum dependence ratio, and proportion of pairwise NAs.

Usage

dependence(mat, cont = NULL, sc.order = 0)
dependence(mat, cont = NULL, sc.order = 0)

Arguments

`mat`	$k$ -column matrix with $n$ observations of a $k$ -dimensional random vector (NA values are allowed).
`cont`	vector of column numbers to consider/coerce as continuous random variables (optional).
`sc.order`	order of subcopula approximation (continuous random variables). If $0$ (default) then maximum order $m = n$ is used. Often $m = 50$ is a good recommended value, higher values demand more computing time.

Details

Each of the random variables in the $k$ -dimensional random vector under consideration may be of any kind (discrete, continuous, or mixed). NA values are allowed.

Value

A 3-dimensional array $k\times k\times 4$ with pairwise monotone and supremum dependence, monotone/supremum dependence ratio, and proportion of pairwise NAs.

Note

NA values are allowed.

Author(s)

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

References

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.

Examples

V <- runif(300)  # Continuous Uniform(0,1)
W <- V*(1-V)     # Continuous transform of V
# X given V=v as continuous Uniform(0,v)
X <- mapply(runif, rep(1, length(V)), rep(0, length(V)), V)
Y <- 1*(0.2 < X)*(X < 0.6)  # Discrete transform of X
Z <- X*(0.1 < X)*(X < 0.9) + 1*(X >= 0.9)  # Mixed transform of X
V[1:10] <- NA  # Introducing some NAs
W[3:12] <- NA  # Introducing some NAs
Y[5:25] <- NA  # Introducing some NAs
vector5D <- cbind(V, W, X, Y, Z)  # Matrix of 5-variate observations
# Monotone and supremum dependence, ratio and proportion of NAs:
(deparray <- dependence(vector5D, cont = c(1, 2, 3), 30))
# Pearson's correlations:
cor(vector5D, method = "pearson", use = "pairwise.complete.obs")
# Spearman's correlations:
cor(vector5D, method = "spearman", use = "pairwise.complete.obs") 
# Kendall's correlations:
cor(vector5D, method = "kendall", use = "pairwise.complete.obs")   
pairs(vector5D)  # Matrix of pairwise scatterplots
V <- runif(300)  # Continuous Uniform(0,1)
W <- V*(1-V)     # Continuous transform of V
# X given V=v as continuous Uniform(0,v)
X <- mapply(runif, rep(1, length(V)), rep(0, length(V)), V)
Y <- 1*(0.2 < X)*(X < 0.6)  # Discrete transform of X
Z <- X*(0.1 < X)*(X < 0.9) + 1*(X >= 0.9)  # Mixed transform of X
V[1:10] <- NA  # Introducing some NAs
W[3:12] <- NA  # Introducing some NAs
Y[5:25] <- NA  # Introducing some NAs
vector5D <- cbind(V, W, X, Y, Z)  # Matrix of 5-variate observations
# Monotone and supremum dependence, ratio and proportion of NAs:
(deparray <- dependence(vector5D, cont = c(1, 2, 3), 30))
# Pearson's correlations:
cor(vector5D, method = "pearson", use = "pairwise.complete.obs")
# Spearman's correlations:
cor(vector5D, method = "spearman", use = "pairwise.complete.obs") 
# Kendall's correlations:
cor(vector5D, method = "kendall", use = "pairwise.complete.obs")   
pairs(vector5D)  # Matrix of pairwise scatterplots

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)
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.

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 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)

Bivariate Empirical Sucopula of Given Approximation Order

Description

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

Usage

subcopemc(mat.xy, m = nrow(mat.xy), display = FALSE)
subcopemc(mat.xy, m = nrow(mat.xy), display = FALSE)

Arguments

`mat.xy`	2-column matrix with bivariate observations of a continuous random vector $(X,Y)$ .
`m`	integer value of approximation order, where $m = 2,...,n$ with $n$ equal to sample size.
`display`	logical value indicating if graphs and dependence values should be displayed.

Details

Both random variables $X$ and $Y$ must be continuous, and therefore repeated values in the sample are not expected. If found, jitter will be applied to break ties. 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.
`order`	numeric value of approximation order.
`std.sample`	2-column matrix with the standardized bivariate sample.
`sample`	2-column matrix with the original bivariate sample of $(X,Y)$ .

Note

If approximation order $m > 2000$ calculation may take more than 2 minutes. Usually $m = 50$ would be enough for an acceptable approximation.

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.

Examples

## Example 1: Independent Normal and Gamma

n <- 300  # sample size
X <- rnorm(n)         # Normal(0,1)
Y <- rgamma(n, 2, 3)  # Gamma(2,3)
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 <- subcopemc(XY,, display = TRUE)
str(SC)
## Approximation of order m = 15
SCm15 <- subcopemc(XY, 15, display = TRUE)
str(SCm15)

## Example 2: Non-monotone dependence

n <- 300  # sample size
Theta <- runif(n, 0, 2*pi)  # Uniform circular distribution
X <- cos(Theta)
Y <- sin(Theta)
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 <- subcopemc(XY,, display = TRUE)
str(SC)
## Approximation of order m = 15
SCm15 <- subcopemc(XY, 15, display = TRUE)
str(SCm15)
## Example 1: Independent Normal and Gamma

n <- 300  # sample size
X <- rnorm(n)         # Normal(0,1)
Y <- rgamma(n, 2, 3)  # Gamma(2,3)
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 <- subcopemc(XY,, display = TRUE)
str(SC)
## Approximation of order m = 15
SCm15 <- subcopemc(XY, 15, display = TRUE)
str(SCm15)

## Example 2: Non-monotone dependence

n <- 300  # sample size
Theta <- runif(n, 0, 2*pi)  # Uniform circular distribution
X <- cos(Theta)
Y <- sin(Theta)
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 <- subcopemc(XY,, display = TRUE)
str(SC)
## Approximation of order m = 15
SCm15 <- subcopemc(XY, 15, display = TRUE)
str(SCm15)

Package 'subcopem2D'

Help Index

Bernstein Copula Approximation

Description

Usage

Arguments

Details

Value

Acknowledgement

Note

Author(s)

References

See Also

Examples

Dependence Measures

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Bivariate Empirical Subcopula

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Bivariate Empirical Sucopula of Given Approximation Order

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples