Package {depcoeff}

coeffpml

Description

Computing whatever

Usage

coeffpml(u1, v1, u2, v2, amin, parp, parh, n, na, mf)

Arguments

Details

Some details

Value

NumericVector

Author(s)

Eckhard Liebscher

Kendall regression coefficient

Description

The function kendr evaluates the Kendall regression coefficient. It describes how well the target variable y can be fitted by a function of the regressor variables which increases or decreases as the regressors increase.

Usage

kendr(x,y,direction=NULL,out=0,outsd=TRUE,eps=0.95)

Arguments

Value

A list of the coefficients for several directions with components:

dcoeff

Kendall regression coefficient

dir

direction vector

Pxx

fraction of X \le \breve{X} where \breve{X} has the same distribution as X

sd

standard error

conf

confidence interval

References

Eckhard Liebscher (2021). Kendall regression coefficient. Computational Statistics and Data Analysis 157 (2021). 107140

Examples

library(MASS)
data <- gilgais
kendr(data[,1:3],data[,4],out=1)

Multivariate Kendall regression coefficient

Description

The function kendrm evaluates the multivariate Kendall regression coefficient. It describes how well the response vector y can be fitted by a function of the regressor variables which increases or decreases as the regressors increases.

Usage

kendrm(x,y,direction=NULL,out=0,outsd=TRUE,eps=0.95)

Arguments

Value

A list of the coefficients for several directions with components:

dcoeff

Kendall regression coefficient

dir

direction vector

Pxx

fraction of X \le \breve{X} where \breve{X} has the same distribution as X

sd

standard deviation

conf

confidence interval

References

Eckhard Liebscher (2026). Kendall regression coefficient revisited.

Examples

library(faraway)
y<- fat[,c("brozek","siri")]
x<- fat[,c("weight","adipos","abdom","hip")]
kendrm(x,y,direction=NULL,out=1,outsd=FALSE)
kendrm(x,y,direction=NULL,out=0,outsd=TRUE)

Kendall regression coefficient for split domains

Description

The function kendrs evaluates the multivariate Kendall regression coefficient for two regressors and split regressor region. It describes how well the response variable can be fitted in each split region by a function which increases or decreases as the regressors increase.

Usage

kendrs(x,y,splitp=NULL)

Arguments

Value

A list of Kendall regression coefficients for the 4 split regions (11=left lower region, 12=left upper region...) with components

dcoeff++

split coefficient ++

dcoeff+-

split coefficient +-

totalcoeff

total coefficient

directions

optimal directions

direction ++ means that y increases whenever both regressors increases direction +- means that y increases whenever the first regressor increases and the other regressor decreases..etc.

References

Eckhard Liebscher (2021). Kendall regression coefficient. Computational Statistics and Data Analysis 157 (2021). 107140

Examples

library(MASS)
data<- gilgais
kendrs(data[,1:2],data[,3],splitp=c(0.4,0.6))

Multivariate Kendall's tau

Description

The function kendtaum evaluates the multivariate Kendall's tau coefficient. It describes the dependence of the variables in the data matrix

Usage

kendtaum(x,outsd=TRUE,eps=0.95)

Arguments

Value

A list of the coefficients for several directions with components:

dcoeff

multivariate Kendall's tau

sd

standard error

conf

confidence interval

References

Schmid, F.; Schmidt, R.; Blumentritt, T.; Gaißer, S.; Ruppert, M. (2010). Copula-based measures of multivariate association. in F. Durante, W. Härdle, P. Jaworski, T. Rychlik (eds.) Copula Theory and Its Applications. Springer Berlin, 2010.

Examples

library(MASS)
data <- gilgais
kendtaum(data[,1:4],outsd=TRUE)

Spearman regression coefficient

Description

The function spearr evaluates the multivariate Spearman regression coefficient. It describes how well the target variable y can be fit by a function of regressor variables which is increasing w.r.t. some regressors and decreasing w.r.t. the other regressors.

Usage

spearr(x,y,direction=NULL,out=0)

Arguments

Value

A list containing

dcoeff

Spearman regression coefficient

dir

direction vector

References

Eckhard Liebscher (2021). On a multivariate version of Spearman's correlation coefficient for regression: Properties and Applications. Asian Journal of Statistical Sciences 1, No. 2, 123-150.

Examples

library(MASS)
data <- gilgais
spearr(data[,1:3],data[,4],out=1)

Spearman regression coefficient for split domains

Description

The function spearrs evaluates the multivariate Spearman regression coefficient for two regressors and split regressor region. It describes how well the response variable can be fitted in each split region by a function which increases or decreases as the regressors increase.

Usage

spearrs(x,y,splitp=NULL)

Arguments

Value

A list of Spearman regression coefficients for the 4 split regions (11=left lower region, 12=left upper region...) with components:

dcoeff++

split coefficient ++

dcoeff+-

split coefficient +-

totalcoeff

total coefficient

directions

optimal directions for the several split regions

direction ++ means that y increases whenever both regressors increases direction +- means that y increases whenever the first regressor increases and the other regressor decreases..etc.

References

Eckhard Liebscher (2021). On a multivariate version of Spearman's correlation coefficient for regression: Properties and Applications. Asian Journal of Statistical Sciences 1, No. 2, 123-150.

Examples

library(MASS)
data<- gilgais
spearrs(data[,1:2],data[,3],splitp=c(0.4,0.6))

Xi dependence coefficient

Description

xic is a function to evaluate the xi dependence coefficient (one interval) of two random variables x and y which is based on the copula. Two specific coefficients are available: the power coefficient and the Huber function coefficient.

Usage

xic(x,y,method="power",methodF=1,parH=0.5,parp=1.5,outsd=TRUE,eps=0.95)

Arguments

Details

Let X_{1},\ldots ,X_{n} be the sample of the X variable. Formulas for the estimators of values F(X_{i}) of the distribution function: methodF = 1 \rightarrow \hat{F}(X_{i})=\frac{1}{n}\textrm{rank}(X_{i}) methodF = 2 \rightarrow \hat{F}^{1}(X_{i})=\frac{1}{n+1}\textrm{rank}(X_{i}) methodF = 3 \rightarrow \hat{F}^{2}(X_{i})=\frac{1}{\sqrt{n^{2}-1}}\textrm{rank}(X_{i}) The values of the distribution function of Y are treated analogously.

Value

A list of the following vectors:

power

zeta coefficient with power function, standard error, confidence interval

Huber

zeta coefficient with Huber function, standard error, confidence interval

The zeta dependence coefficient of two random variables is bounded by 1. The higher the value the stronger is the dependence.

References

Eckhard Liebscher (2014). Copula-based dependence measures. Dependence Modeling 2 (2014), 49-64

Examples

library(MASS)
data<- gilgais
xic(data[,1],data[,2])

Zeta dependence coefficient

Description

zetac is a function to evaluate the zeta dependence coefficient (one interval) of two random variables x and y which is based on the copula. Four specific coefficients are available: the Spearman coefficient, Spearman's footrule, the power coefficient and the Huber function coefficient.

Usage

zetac(x,y,method="Spearman",methodF=1,parH=0.5,parp=1.5,outsd=TRUE,eps=0.95)

Arguments

Details

Let X_{1},\ldots ,X_{n} be the sample of the X variable. Formulas for the estimators of values F(X_{i}) of the distribution function: methodF = 1 \rightarrow \hat{F}(X_{i})=\frac{1}{n}\textrm{rank}(X_{i}) methodF = 2 \rightarrow \hat{F}^{1}(X_{i})=\frac{1}{n+1}\textrm{rank}(X_{i}) methodF = 3 \rightarrow \hat{F}^{2}(X_{i})=\frac{1}{\sqrt{n^{2}-1}}\textrm{rank}(X_{i}) The values of the distribution function of Y are treated analogously.

Value

A list of the following vectors:

Spearman

zeta Spearman coefficient, standard error, confidence interval

footrule

zeta footrule coefficient, standard error, confidence interval

power

zeta coefficient with power function, standard error, confidence interval

Huber

zeta coefficient with Huber function, standard error, confidence interval

The zeta dependence coefficient of two random variables is bounded by 1. The higher the value the stronger is the dependence.

References

Eckhard Liebscher (2014). Copula-based dependence measures. Dependence Modeling 2 (2014), 49-64

Examples

library(MASS)
data<- gilgais
zetac(data[,1],data[,2],method="all")

Zeta coefficient of piecewise monotonicity with split domain

Description

The function zetaci evaluates the coefficient of piecewise monotonicity of variables x and y where the x-domain is split into a fixed number of intervals.

Usage

zetaci(x,y,a,method="Spearman",methodF=1,parH=0.5,parp=1.5)

Arguments

Details

Let X_{1},\ldots ,X_{n} be the sample of the X variable. Formulas for the estimators of values F(X_{i}) of the distribution function: methodF = 1 \rightarrow \hat{F}(X_{i})=\frac{1}{n}\textrm{rank}(X_{i}) methodF = 2 \rightarrow \hat{F}^{1}(X_{i})=\frac{1}{n+1}\textrm{rank}(X_{i}) methodF = 3 \rightarrow \hat{F}^{2}(X_{i})=\frac{1}{\sqrt{n^{2}-1}}\textrm{rank}(X_{i}) The values of the distribution function of Y are treated analogously.

Value

A list of zeta dependence coefficients of piecewise monotonicity of two random variables containing the following elements:

Spearman

zeta Spearman plusminus/minusplus coefficient

footrule

zeta Spearman's footrule plusminus/minusplus coefficient

power

zeta power plusminus/minusplus coefficient

Huber

zeta Huber plusminus/minusplus coefficient

References

Eckhard Liebscher (2017). Copula-based dependence measures for piecewise monotonicity. Dependence Modeling 5 (2017), 198-220

Examples

library(MASS)
x<- seq(0,1.0,by=0.01)
eps<- rnorm(length(x),sd=0.05)
y<- x-2*(x>=0.4)*(x-0.4)+4*(x>=0.75)*(x-0.75)+eps
zetaci(x,y, a=c(0.25, 0.5, 0.75))

Zeta dependence coefficient of piecewise monotonicity

Description

zetapm is a function to evaluate the zeta dependence coefficients of piecewise monotonicity of two random variables x and y which is based on the copula. The regressor domain (domain of x) is split into two parts. The function searches for the optimal splitting point to obtain maximum depedence. The main part of the function is coded as C++ procedure

Usage

zetapm(x,y,amin=0.25,method="all",methodF=1,parp=1.5,parH=0.5)

Arguments

Details

Let X_{1},\ldots ,X_{n} be the sample of the X variable. Formulas for the estimators of values F(X_{i}) of the distribution function: methodF = 1 \rightarrow \hat{F}(X_{i})=\frac{1}{n}\textrm{rank}(X_{i}) methodF = 2 \rightarrow \hat{F}^{1}(X_{i})=\frac{1}{n+1}\textrm{rank}(X_{i}) methodF = 3 \rightarrow \hat{F}^{2}(X_{i})=\frac{1}{\sqrt{n^{2}-1}}\textrm{rank}(X_{i}) The values of the distribution function of Y are treated analogously.

Value

A list of zeta dependence coefficients of piecewise monotonicity of two random variables containing the following elements:

plusminuscoeff

zeta plusminus coefficients: Spearman, footrule, power, Huber coefficients if chosen

position1

positions of the optimal split points within the x-value (zeta plusminus coefficients)

minuspluscoeff

zeta minuspluscoeff coefficients: Spearman, footrule, power, Huber coefficients if chosen

position2

positions of the optimal split points within the x-value (zeta minusplus coefficients)

References

Eckhard Liebscher (2017). Copula-based dependence measures for piecewise monotonicity. Dependence Modeling 5 (2017), 198-220

Examples

x<- seq(0,1.0,by=0.01)
eps<- rnorm(length(x),sd=0.05)
y<- x-2*(x>=0.4)*(x-0.4)+eps
zetapm(x,y,amin=0.2,method="all",methodF=1,parp=1.5,parH=0.5)