Package 'kdecopula' reference manual

Title:	Kernel Smoothing for Bivariate Copula Densities
Description:	Provides fast implementations of kernel smoothing techniques for bivariate copula densities, in particular density estimation and resampling, see Nagler (2018) <doi:10.18637/jss.v084.i07>.
Authors:	Thomas Nagler [aut, cre], Kuangyu Wen [ctb]
Maintainer:	Thomas Nagler <[email protected]>
License:	GPL-3
Version:	0.9.2
Built:	2024-11-03 06:39:01 UTC
Source:	CRAN

Bandwidth selection for the Bernstein copula estimator

Description

The optimal size of knots is chosen by a rule of thumb adapted from Rose (2015).

Usage

bw_bern(udata)
bw_bern(udata)

Arguments

udata

data.

Details

The formula is

$max(1, round(n^(1/3) * exp(abs(rho)^(1/n)) * (abs(rho) + 0.1))),$

where $\rho$ is the empirical Spearman's rho of the data.

Value

optimal order of the Bernstein polynomials.

Bandwidth selection for the beta kernel estimator

Description

The bandwidth is selected by minimizing the MISE using the Frank copula as the reference family. The copula parameter is set by inversion of Kendall's tau. See Nagler (2014) for details.

Usage

bw_beta(udata)
bw_beta(udata)

Arguments

udata

data.

Details

To speed things up, optimal bandwidths have been pre-calculated on a grid of tau values.

Value

A scalar bandwidth parameter.

References

Nagler, T. (2014). Kernel Methods for Vine Copula Estimation. Master's Thesis, Technische Universitaet Muenchen, https://mediatum.ub.tum.de/node?id=1231221

Bandwidth selection for the mirror-reflection estimator

Description

The bandwidth is selected by minimizing the MISE using the Frank copula as the reference family. The copula parameter is set by inversion of Kendall's tau. See Nagler (2014) for details.

Usage

bw_mr(udata)
bw_mr(udata)

Arguments

udata

data.

Details

To speed things up, optimal bandwidths have been pre-calculated on a grid of tau values.

Value

A scalar bandwidth parameter.

References

Nagler, T. (2014). Kernel Methods for Vine Copula Estimation. Master's Thesis, Technische Universitaet Muenchen, https://mediatum.ub.tum.de/node?id=1231221

Bandwidth selection for the transformation kernel estimator

Description

The bandwidth is selected by a rule of thumb. It approximately minimizes the MISE of the Gaussian copula on the transformed domain. The usual normal reference matrix is multiplied by 1.25 to account for the higher variance on the copula level.

Usage

bw_t(udata)
bw_t(udata)

Arguments

udata

data.

Details

The formula is

$1.25 n^{-1 / 6} \hat{\Sigma}^{1/2},$

where $\hat{Sigma}$ is empirical covariance matrix of the transformed random vector.

Value

A 2 x 2 bandwidth matrix.

References

Nagler, T. (2014). Kernel Methods for Vine Copula Estimation. Master's Thesis, Technische Universitaet Muenchen, https://mediatum.ub.tum.de/node?id=1231221

Bandwidth selection for the transformation local likelihood estimator

Description

The bandwidth is selected by a rule of thumb similar to bw_t.

Usage

bw_tll(udata, deg)
bw_tll(udata, deg)

Arguments

`udata`	data.
`deg`	degree of the polynomial.

Details

The formula is

$5 n^{-1 / (4q + 2)} \hat{\Sigma}^{1/2},$

where $\hat{Sigma}$ is empirical covariance matrix of the transformed random vector and $q = 1$ for TLL1 and $q = 2$ for TLL2.

Value

A 2 x 2 bandwidth matrix.

Nearest-neighbor bandwidth selection for the transformation local likelihood estimator

Description

The smoothing parameters is selected by the method of Geenens et al. (2017). It uses principal components for the rotation matrix and selects the nearest neighbor fraction along each principal direction by approximate least-squares cross-validation.

Usage

bw_tll_nn(udata, deg)
bw_tll_nn(udata, deg)

Arguments

`udata`	data.
`deg`	degree of the polynomial.

Value

A list with entries:

B: rotation matrix,
alpha: nearest neighbor fraction (this one is multiplied with mult in kdecop()),
kappa: correction factor for differences in roughness along the axes,

see Geenens et al. (2017).

References

Geenens, G., Charpentier, A., and Paindaveine, D. (2017). Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli, 23(3), 1848-1873.

Nearest-neighbor bandwidth selection for the tapered transformation estimator

Description

The smoothing parameters are selected by the method of Wen and Wu (2015).

Usage

bw_tt_pi(udata, rho.add = TRUE)

bw_tt_cv(udata, rho.add = T)
bw_tt_pi(udata, rho.add = TRUE)

bw_tt_cv(udata, rho.add = T)

Arguments

`udata`	data.
`rho.add`	logical; whether a rotation (correlation) parameter shall be included.

Value

Optimal smoothing parameters as in Wen and Wu (2015): a numeric vector of length 4; entries are $(h, \rho, \theta_1, \theta_2)$ .

Author(s)

Kuangyu Wen

References

Wen, K. and Wu, X. (2015). Transformation-Kernel Estimation of the Copula Density, Working paper, http://agecon2.tamu.edu/people/faculty/wu-ximing/agecon2/public/copula.pdf

Dependence measures of a `kdecop()` fit

Description

Calculates several dependence measures derived from the copula density. All measures except "blomqvist" are computed by quasi Monte Carlo methods (see rkdecop().

Usage

dep_measures(object, measures = "all", n_qmc = 10^3, seed = 5)
dep_measures(object, measures = "all", n_qmc = 10^3, seed = 5)

Arguments

`object`	an object of class `kdecopula`.
`measures`	which measures to compute, see Details.
`n_qmc`	the number of quasi Monte Carlo samples.
`seed`	the seed for quasi Monte Carlo integration.

Value

A named vector of dependence measures.

The following measures are available:

"kendall": Kendall's $\tau$ , see Nelsen (2007); computed as the sample version of a quasi Monte Carlo sample.
"spearman": Spearman's $\rho$ , see Nelsen (2007); computed as the sample version of a quasi Monte Carlo sample.
"blomqvist": Blomqvist's $\beta$ , see Nelsen (2007); computed as $4C(0.5, 0.5) - 1$ .
"gini": Gini's $\gamma$ , see Nelsen (2007); computed by quasi Monte Carlo integration.
"vd_waerden": van der Waerden's coefficient, see Genest and Verret (2005); computed as the sample version of a quasi Monte Carlo sample.
"minfo": mutual information, see Joe (1989); computed by quasi Monte Carlo integration.
"linfoot": Linfoot's correlation coefficient, see Joe (1989); computed by quasi Monte Carlo integration.

References

Nelsen, R. (2007). An introduction to copulas. Springer Science & Business Media, 2007.

Genest, C., and Verret, F. (2005). Locally most powerful rank tests of independence for copula models. Journal of Nonparametric Statistics, 17(5)

Joe, H. (1989). Relative Entropy Measures of Multivariate Dependence. Journal of the American Statistical Association, 84(405)

Examples

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimate copula density and calculate dependence measures
fit <- kdecop(udat[, 5:6])
dep_measures(fit)

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimate copula density and calculate dependence measures
fit <- kdecop(udat[, 5:6])
dep_measures(fit)

Working with `kdecopula` objects

Description

The function kdecop() stores it's result in object of class kdecopula. The density estimate can be evaluated on arbitrary points with dkdecop(); the cdf with pkdecop(). Furthermore, synthetic data can be simulated with rkdecop().

Usage

dkdecop(u, obj, stable = FALSE)

pkdecop(u, obj)

rkdecop(n, obj, quasi = FALSE)
dkdecop(u, obj, stable = FALSE)

pkdecop(u, obj)

rkdecop(n, obj, quasi = FALSE)

Arguments

`u`	`mx2` matrix of evaluation points.
`obj`	`kdecopula` object.
`stable`	logical; option for stabilizing the estimator: the estimated density is cut off at $50$ .
`n`	integer; number of observations.
`quasi`	logical; the default (`FALSE`) returns pseudo-random numbers, use `TRUE` for quasi-random numbers (generalized Halton, see `qrng::ghalton()`).

Value

A numeric vector of the density/cdf or a n x 2 matrix of simulated data.

Author(s)

Thomas Nagler

References

#' Nagler, T. (2018) kdecopula: An R Package for the Kernel Estimation of Bivariate Copula Densities. Journal of Statistical Software 84(7), 1-22

#' Geenens, G., Charpentier, A., and Paindaveine, D. (2017). Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli, 23(3), 1848-1873.

Nagler, T. (2014). Kernel Methods for Vine Copula Estimation. Master's Thesis, Technische Universitaet Muenchen, https://mediatum.ub.tum.de/node?id=1231221

Cambou, T., Hofert, M., Lemieux, C. (2015). A primer on quasi-random numbers for copula models, arXiv:1508.03483

Examples


## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimation of copula density of variables 5 and 6
fit <- kdecop(udat[, 5:6])
plot(fit) 

## evaluate density estimate at (u1,u2)=(0.123,0.321)
dkdecop(c(0.123, 0.321), fit) 

## evaluate cdf estimate at (u1,u2)=(0.123,0.321)
pkdecop(c(0.123, 0.321), fit) 

## simulate 500 samples from density estimate
plot(rkdecop(500, fit))

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimation of copula density of variables 5 and 6
fit <- kdecop(udat[, 5:6])
plot(fit) 

## evaluate density estimate at (u1,u2)=(0.123,0.321)
dkdecop(c(0.123, 0.321), fit) 

## evaluate cdf estimate at (u1,u2)=(0.123,0.321)
pkdecop(c(0.123, 0.321), fit) 

## simulate 500 samples from density estimate
plot(rkdecop(500, fit))

Extract fitted values from a `kdecop()` fits.

Description

Simply calls predict(object, object$udata, what).

Usage

## S3 method for class 'kdecopula'
fitted(object, what = "pdf", ...)
## S3 method for class 'kdecopula'
fitted(object, what = "pdf", ...)

Arguments

`object`	an object of class `kdecopula`.
`what`	what to predict, one of `c("pdf", "cdf", "hfunc1", "hfunc2", "hinv1", "hinv2")`.
`...`	unused.

Examples

data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))
fit <- kdecop(udat[, 5:6])

all.equal(fitted(fit), predict(fit, fit$udata))
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))
fit <- kdecop(udat[, 5:6])

all.equal(fitted(fit), predict(fit, fit$udata))

H-function and inverse of a `kdecop()` fit

Description

Evaluates the h-function (or its inverse) corresponding to a kdecopula object. H-functions are conditional distribution functions obtained by integrating the copula density w.r.t. to one of its arguments (see also VineCopula::BiCopHfunc().

Usage

hkdecop(u, obj, cond.var, inverse = FALSE)
hkdecop(u, obj, cond.var, inverse = FALSE)

Arguments

`u`	$n x 2$ matrix of evaluation points.
`obj`	`kdecopula` object.
`cond.var`	integer; `cond.var = 1` conditions on the first variable, `cond.var = 2` on the second.
`inverse`	logical; indicates whether the h-function or its inverse shall be calculated.

Value

A length $n$ vector of the (inverse) h-function evaluated at u.

Author(s)

Thomas Nagler

Examples

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimation of copula density of variables 5 and 6
fit <- kdecop(udat[, 5:6])
plot(fit) 

## evaluate h-function estimate and its inverse at (u1|u2) = (0.123 | 0.321)
hkdecop(c(0.123, 0.321), fit, cond.var = 2) 
hkdecop(c(0.123, 0.321), fit, cond.var = 2, inverse = TRUE) 

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimation of copula density of variables 5 and 6
fit <- kdecop(udat[, 5:6])
plot(fit) 

## evaluate h-function estimate and its inverse at (u1|u2) = (0.123 | 0.321)
hkdecop(c(0.123, 0.321), fit, cond.var = 2) 
hkdecop(c(0.123, 0.321), fit, cond.var = 2, inverse = TRUE)

Bivariate kernel copula density estimation

Description

Based on samples from a bivariate copula, the copula density is estimated. The user can choose between different methods. If no bandwidth is provided by the user, it will be set by a method-specific automatic selection procedure. The related (d/p/r)kdecop functions evaluate the density and cdf or simulate synthetic data, respectively.

Usage

kdecop(udata, bw = NA, mult = 1, method = "TLL2nn", knots = 30,
  renorm.iter = 3L, info = TRUE)
kdecop(udata, bw = NA, mult = 1, method = "TLL2nn", knots = 30,
  renorm.iter = 3L, info = TRUE)

Arguments

`udata`	`nx2` matrix of copula data.
`bw`	bandwidth specification; if `NA`, `bw` is selected automatically (see Details); Otherwise, please provide `"T", "TLL1", "TLL2"`: a $2x2$ bandwidth matrix, `"TLL1nn", "TLL2nn"`: a list with (named) entries `B`, `alpha`, and `kappa`, `"TTCV", "TTPI"`: a numeric vector of length four containing $(h, \rho, \theta_1, \theta_2)$ , c.f. Wen and Wu (2015), `"MR", "beta"`: a positive real number.
`mult`	bandwidth multiplier, has to be positive; useful for making estimates more/less smooth manually.
`method`	`"T"`: transformation estimator based on classical bivariate kernel estimation (e.g., Geenenens et al., 2014), `"TLL1"`: transformation estimator with log-linear local likelihood estimation (Geenenens et al., 2014), `"TLL2"`: transformation estimator with log-quadratic local likelihood estimation (Geenenens et al., 2014), `"TLL1nn"`: transformation estimator with log-linear local likelihood estimation and nearest-neighbor bandwidths (Geenenens et al., 2014), `"TLL2nn"`: transformation estimator with log-quadratic local likelihood estimation and nearest-neighbor bandwidths (Geenenens et al., 2014), `"TTPI"`: tapered transformation estimator with plug-in bandwidths (Wu and Wen, 2015), `"TTCV"`: tapered transformation estimator with profile cross-validation bandwidths (Wu and Wen, 2015), `"MR"`: mirror-reflection estimator (Gijbels and Mielniczuk, 1990), `"beta"`: beta kernel estimator (Charpentier et al., 2006), `"bern"`: Bernstein copula estimator (Sanchetta and Satchell, 2004); the coefficients are adjusted by the method of Weiss and Scheffer (2012).
`knots`	integer; number of knots in each dimension for the spline approximation.
`renorm.iter`	integer; number of iterations for the renormalization procedure (see Details).
`info`	logical; if `TRUE`, additional information about the estimate will be gathered (see Value).

Details

We use a Gaussian product kernel function for all methods except the beta kernel and Bernstein estimators. For details on bandwidth selection for a specific method, see: bw_t, bw_tll, bw_tll_nn, bw_tt_pi, bw_tt_cv, bw_mr, bw_beta, bw_bern.

Kernel estimates are usually no proper copula densities. In particular, the estimated marginal densities are not uniform. We mitigate this issue by a renormalization procedure. The number of iterations of the renormalization algorithm can be specified with the renorm.iter argument. Typically, a very small number of iterations is sufficient.

Value

The function kdecop returns an object of class kdecopula that contains all information necessary for evaluation of the estimator. If no bandwidth was provided in the function call, the automatically selected value can be found in the variable object$bw. If info=TRUE, also the following will be available under object$info:

`likvalues`	Estimator evaluated in sample points
`loglik`	Log likelihood
`effp`	Effective number of parameters
`AIC`	Akaike information criterion
`cAIC`	Bias-corrected version of Akaike information criterion
`BIC`	Bayesian information criterion.

The density estimate can be evaluated on arbitrary points with dkdecop; the cdf with pkdecop. Furthermore, synthetic data can be simulated with rkdecop, and several plotting options are available with plot and contour.

Note

The implementation of the tapered transformation estimator ("TTPI"/"TTCV") was kindly provided by Kuangyu Wen.

Author(s)

Thomas Nagler

References

Nagler, T. (2018) kdecopula: An R Package for the Kernel Estimation of Bivariate Copula Densities. Journal of Statistical Software 84(7), 1-22

Geenens, G., Charpentier, A., and Paindaveine, D. (2017). Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli, 23(3), 1848-1873.

Wen, K. and Wu, X. (2015). Transformation-Kernel Estimation of the Copula Density, Working paper, http://agecon2.tamu.edu/people/faculty/wu-ximing/agecon2/public/copula.pdf

Gijbels, I. and Mielniczuk, J. (1990). Estimating the density of a copula function. Communications in Statistics - Theory and Methods, 19(2):445-464.

Charpentier, A., Fermanian, J.-D., and Scaillet, O. (2006). The estimation of copulas: Theory and practice. In Rank, J., editor, Copulas: From theory to application in finance. Risk Books.

Weiss, G. and Scheffer, M. (2012). Smooth Nonparametric Bernstein Vine Copulas. arXiv:1210.2043

Nagler, T. (2014). Kernel Methods for Vine Copula Estimation. Master's Thesis, Technische Universitaet Muenchen, https://mediatum.ub.tum.de/node?id=1231221

Examples


## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimation of copula density of variables 5 and 6
fit <- kdecop(udat[, 5:6])
summary(fit)
plot(fit)
contour(fit)

## evaluate density estimate at (u1,u2)=(0.123,0.321)
dkdecop(c(0.123, 0.321), fit)

## evaluate cdf estimate at (u1,u2)=(0.123,0.321)
pkdecop(c(0.123, 0.321), fit)

## simulate 500 samples from density estimate
plot(rkdecop(500, fit))  # pseudo-random
plot(rkdecop(500, fit, quasi = TRUE))  # quasi-random


## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimation of copula density of variables 5 and 6
fit <- kdecop(udat[, 5:6])
summary(fit)
plot(fit)
contour(fit)

## evaluate density estimate at (u1,u2)=(0.123,0.321)
dkdecop(c(0.123, 0.321), fit)

## evaluate cdf estimate at (u1,u2)=(0.123,0.321)
pkdecop(c(0.123, 0.321), fit)

## simulate 500 samples from density estimate
plot(rkdecop(500, fit))  # pseudo-random
plot(rkdecop(500, fit, quasi = TRUE))  # quasi-random

Kernel Smoothing for Bivariate Copula Densities

Description

This package provides fast implementations of kernel estimators for the copula density. Due to its several plotting options it is particularly useful for the exploratory analysis of dependence structures. It can be further used for flexible nonparametric estimation of copula densities and resampling.

Details

The function kdecop can be used to estimate a copula density with a number of popular kernel estimators. The density estimate can be evaluated on arbitrary points with dkdecop; the cdf with pkdecop. Furthermore, synthetic data can be simulated with rkdecop, and several plot options are provided by plot.kdecopula.

Author(s)

Thomas Nagler

References

Nagler, T. (2018) kdecopula: An R Package for the Kernel Estimation of Bivariate Copula Densities. Journal of Statistical Software 84(7), 1-22

Gijbels, I. and Mielniczuk, J. (1990). Estimating the density of a copula function. Communications in Statistics - Theory and Methods, 19(2):445-464.

Charpentier, A., Fermanian, J.-D., and Scaillet, O. (2006). The estimation of copulas: Theory and practice. In Rank, J., editor, Copulas: From theory to application in finance. Risk Books.

Geenens, G., Charpentier, A., and Paindaveine, D. (2017). Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli, 23(3), 1848-1873.

Nagler, T. (2014). Kernel Methods for Vine Copula Estimation. Master's Thesis, Technische Universitaet Muenchen, https://mediatum.ub.tum.de/node?id=1231221

Wen, K. and Wu, X. (2015). Transformation-Kernel Estimation of the Copula Density, Working paper, http://agecon2.tamu.edu/people/faculty/wu-ximing/agecon2/public/copula.pdf

Examples


## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x)/(length(x)+1))

## estimation of copula density of variables 5 and 6
dens.est <- kdecop(udat[, 5:6])
plot(dens.est)

## evaluate density estimate at (u1,u2)=(0.123,0.321)
dkdecop(c(0.123, 0.321), dens.est)

## evaluate cdf estimate at (u1,u2)=(0.123,0.321)
pkdecop(c(0.123, 0.321), dens.est)

## simulate 500 samples from density estimate
rkdecop(500, dens.est)

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x)/(length(x)+1))

## estimation of copula density of variables 5 and 6
dens.est <- kdecop(udat[, 5:6])
plot(dens.est)

## evaluate density estimate at (u1,u2)=(0.123,0.321)
dkdecop(c(0.123, 0.321), dens.est)

## evaluate cdf estimate at (u1,u2)=(0.123,0.321)
pkdecop(c(0.123, 0.321), dens.est)

## simulate 500 samples from density estimate
rkdecop(500, dens.est)

Log-Likelihood of a `kdecopula` object

Description

Log-Likelihood of a kdecopula object

Usage

## S3 method for class 'kdecopula'
logLik(object, ...)
## S3 method for class 'kdecopula'
logLik(object, ...)

Arguments

`object`	an object of class `kdecopula`.
`...`	not used.

Value

Returns an object of class logLik containing the log- likelihood, number of observations and effective number of parameters ("df").

Author(s)

Thomas Nagler

Examples

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimation of copula density of variables 5 and 6
fit <- kdecop(udat[, 5:6])

## compute fit statistics
logLik(fit)
AIC(fit)
BIC(fit)

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))

## estimation of copula density of variables 5 and 6
fit <- kdecop(udat[, 5:6])

## compute fit statistics
logLik(fit)
AIC(fit)
BIC(fit)

Plotting `kdecopula` objects

Description

Produces perspective or contour plots for a kdecopula object.

Usage

## S3 method for class 'kdecopula'
plot(x, type = "surface", margins, size, ...)

## S3 method for class 'kdecopula'
contour(x, margins = "norm", size = 100L, ...)
## S3 method for class 'kdecopula'
plot(x, type = "surface", margins, size, ...)

## S3 method for class 'kdecopula'
contour(x, margins = "norm", size = 100L, ...)

Arguments

`x`	`kdecopula` object.
`type`	plot type; either `"contour"` or `"surface"` (partial matching is activated) for a contour or perspective/surface plot respectively.
`margins`	`"unif"` for the original copula density, `"norm"` for the transformed density with standard normal margins, `"exp"` with standard exponential margins, and `"flexp"` with flipped exponential margins. Default is `"norm"` for `type = "contour"`, and `"unif"` for `type = "surface"`.
`size`	integer; the plot is based on values on a $size x size$ grid; default is 100 for `type = "contour"`, and 25 for `type = "surface"`.
`...`	optional arguments passed to `contour` or `wireframe`.

Author(s)

Thomas Nagler

References

Nagler, T. (2018) kdecopula: An R Package for the Kernel Estimation of Bivariate Copula Densities. Journal of Statistical Software 84(7), 1-22

Examples


## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x)/(length(x)+1))

## estimation of copula density of variables 5 and 6
obj <- kdecop(udat[, 5:6])

## plots
plot(obj)  # surface plot of copula density
contour(obj)  # contour plot with standard normal margins
contour(obj, margins = "unif")  # contour plot of copula density

## load data and transform with empirical cdf
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x)/(length(x)+1))

## estimation of copula density of variables 5 and 6
obj <- kdecop(udat[, 5:6])

## plots
plot(obj)  # surface plot of copula density
contour(obj)  # contour plot with standard normal margins
contour(obj, margins = "unif")  # contour plot of copula density

Prediction method for `kdecop()` fits

Description

Predicts the pdf, cdf, or (inverse) h-functions by calling dkdecop(), pkdecop(), or hkdecop().

Usage

## S3 method for class 'kdecopula'
predict(object, newdata, what = "pdf", stable = FALSE,
  ...)
## S3 method for class 'kdecopula'
predict(object, newdata, what = "pdf", stable = FALSE,
  ...)

Arguments

`object`	an object of class `kdecopula`.
`newdata`	evaluation point(s), a length two vector or a matrix with two columns.
`what`	what to predict, one of `c("pdf", "cdf", "hfunc1", "hfunc2", "hinv1", "hinv2")`.
`stable`	only used for `what = "pdf"`, see `dkdecop()`.
`...`	unused.

Value

A numeric vector of predictions.

Examples

data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))
fit <- kdecop(udat[, 5:6])

all.equal(predict(fit, c(0.1, 0.2)), dkdecop(c(0.1, 0.2), fit))
all.equal(predict(fit, udat, "hfunc1"), hkdecop(udat, fit, cond.var = 1))
data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))
fit <- kdecop(udat[, 5:6])

all.equal(predict(fit, c(0.1, 0.2)), dkdecop(c(0.1, 0.2), fit))
all.equal(predict(fit, udat, "hfunc1"), hkdecop(udat, fit, cond.var = 1))

Simulate data from a `kdecop()` fit.

Description

See rkdecop().

Usage

## S3 method for class 'kdecopula'
simulate(object, nsim = 1, seed = NULL, quasi = FALSE,
  ...)
## S3 method for class 'kdecopula'
simulate(object, nsim = 1, seed = NULL, quasi = FALSE,
  ...)

Arguments

`object`	an object of class `kdecopula`.
`nsim`	integer; number of observations.
`seed`	integer; `set.seed(seed)` will be called prior to `rkdecop()`.
`quasi`	logical; the default (`FALSE`) returns pseudo-random numbers, use `TRUE` for quasi-random numbers (generalized Halton, see
`...`	unused.

Value

Simulated data from the fitted kdecopula model.

Examples

data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))
fit <- kdecop(udat[, 5:6])
plot(simulate(fit, 500))

data(wdbc)
udat <- apply(wdbc[, -1], 2, function(x) rank(x) / (length(x) + 1))
fit <- kdecop(udat[, 5:6])
plot(simulate(fit, 500))

Wisconsin Diagnostic Breast Cancer (WDBC)

Description

The data contain measurements on cells in suspicious lumps in a woman's breast. Features are computed from a digitized image of a fine needle aspirate (FNA) of a breast mass. They describe characteristics of the cell nuclei present in the image. All samples are classified as either benign or malignant.

Format

wdbc is a data.frame with 31 columns. The first column indicates whether the sample is classified as benign (B) or malignant (M). The remaining columns contain measurements for 30 features.

Details

Ten real-valued features are computed for each cell nucleus:

a) radius (mean of distances from center to points on the perimeter)
b) texture (standard deviation of gray-scale values)
c) perimeter
d) area
e) smoothness (local variation in radius lengths)
f) compactness (perimeter^2 / area - 1.0)
g) concavity (severity of concave portions of the contour)
h) concave points (number of concave portions of the contour)
i) symmetry
j) fractal dimension ("coastline approximation" - 1)

The references listed below contain detailed descriptions of how these features are computed.

The mean, standard error, and "worst" or largest (mean of the three largest values) of these features were computed for each image, resulting in 30 features.

Note

This breast cancer database was obtained from the University of Wisconsin Hospitals, Madison from Dr. William H. Wolberg.

Source

http://mlr.cs.umass.edu/ml/datasets/Breast+Cancer+Wisconsin+(Diagnostic)

Bache, K. & Lichman, M. (2013). UCI Machine Learning Repository. Irvine, CA: University of California, School of Information and Computer Science.

References

O. L. Mangasarian and W. H. Wolberg: "Cancer diagnosis via linear programming",
SIAM News, Volume 23, Number 5, September 1990, pp 1 & 18.

William H. Wolberg and O.L. Mangasarian: "Multisurface method of pattern separation for medical diagnosis applied to breast cytology",
Proceedings of the National Academy of Sciences, U.S.A., Volume 87, December 1990, pp 9193-9196.

K. P. Bennett & O. L. Mangasarian: "Robust linear programming discrimination of two linearly inseparable sets",
Optimization Methods and Software 1, 1992, 23-34 (Gordon & Breach Science Publishers).

Examples


data(wdbc)
str(wdbc)

data(wdbc)
str(wdbc)

Package 'kdecopula'

Help Index

Bandwidth selection for the Bernstein copula estimator

Description

Usage

Arguments

Details

Value

Bandwidth selection for the beta kernel estimator

Description

Usage

Arguments

Details

Value

References

Bandwidth selection for the mirror-reflection estimator

Description

Usage

Arguments

Details

Value

References

Bandwidth selection for the transformation kernel estimator

Description

Usage

Arguments

Details

Value

References

Bandwidth selection for the transformation local likelihood estimator

Description

Usage

Arguments

Details

Value

Nearest-neighbor bandwidth selection for the transformation local likelihood estimator

Description

Usage

Arguments

Value

References

Nearest-neighbor bandwidth selection for the tapered transformation estimator

Description

Usage

Arguments

Value

Author(s)

References

Dependence measures of a kdecop() fit

Description

Usage

Arguments

Value

References

Examples

Working with kdecopula objects

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Extract fitted values from a kdecop() fits.

Description

Usage

Arguments

See Also

Examples

H-function and inverse of a kdecop() fit

Description

Usage

Arguments

Value

Author(s)

Examples

Bivariate kernel copula density estimation

Description

Usage

Dependence measures of a `kdecop()` fit

Working with `kdecopula` objects

Extract fitted values from a `kdecop()` fits.

H-function and inverse of a `kdecop()` fit

Log-Likelihood of a `kdecopula` object

Plotting `kdecopula` objects

Prediction method for `kdecop()` fits

Simulate data from a `kdecop()` fit.