Kernel Smoothing for Bivariate Copula Densities

Description

This package implements a vine copula based kernel density estimator. The estimator does not suffer from the curse of dimensionality and is therefore well suited for high-dimensional applications (see, Nagler and Czado, 2016).

Details

The multivariate kernel density estimators is implemented by the kdevine function. It combines a kernel density estimator for the margins (kde1d) and a kernel estimator of the vine copula density (kdevinecop). The package is built on top of the copula density estimators in the kdecopula::kdecopula-package and let's you choose from all its implemented methods. Optionally, the vine copula can be estimated parameterically (only the margins are nonparametric).

Author(s)

Thomas Nagler

References

Nagler, T., Czado, C. (2016)
Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas.
Journal of Multivariate Analysis 151, 69-89 (doi:10.1016/j.jmva.2016.07.003)

Nagler, T., Schellhase, C. and Czado, C. (2017)
Nonparametric estimation of simplified vine copula models: comparison of methods arXiv:1701.00845

Nagler, T. (2017)
A generic approach to nonparametric function estimation with mixed data.
arXiv:1704.07457

See Also

Useful links:

• https://github.com/tnagler/kdevine

• Report bugs at https://github.com/tnagler/kdevine/issues

Contour plots of pair copula kernel estimates

Description

Contour plots of pair copula kernel estimates

Usage

## S3 method for class 'kdevinecop'
contour(x, tree = "ALL", xylim = NULL, cex.nums = 1, ...)

Arguments

Examples

data(wdbc, package = "kdecopula")                     # load data
u <- VineCopula::pobs(wdbc[, 5:7], ties = "average")  # rank-transform

# estimate density
fit <- kdevinecop(u)

# contour matrix
contour(fit)

Working with a kde1d object

Description

The density, cdf, or quantile function of a kernel density estimate are evaluated at arbitrary points with dkde1d, pkde1d, and qkde1d respectively.

Usage

dkde1d(x, obj)

pkde1d(x, obj)

qkde1d(x, obj)

rkde1d(n, obj, quasi = FALSE)

Arguments

Value

The density or cdf estimate evaluated at x.

See Also

kde1d

Examples

data(wdbc)  # load data
fit <- kde1d(wdbc[, 5])  # estimate density
dkde1d(1000, fit)        # evaluate density estimate
pkde1d(1000, fit)        # evaluate corresponding cdf
qkde1d(0.5, fit)         # quantile function
hist(rkde1d(100, fit))   # simulate

Evaluate the density of a kdevine object

Description

Evaluate the density of a kdevine object

Usage

dkdevine(x, obj)

Arguments

Value

The density estimate evaluated at x.

See Also

kdevine

Examples

# load data
data(wdbc)

# estimate density (use xmin to indicate positive support)
fit <- kdevine(wdbc[, 5:7], xmin = rep(0, 3))

# evaluate density estimate
dkdevine(c(1000, 0.1, 0.1), fit)

Working with a kdevinecop object

Description

A vine copula density estimate (stored in a kdevinecop object) can be evaluated on arbitrary points with dkevinecop. Furthermore, you can simulate from the estimated density with rkdevinecop.

Usage

dkdevinecop(u, obj, stable = FALSE)

rkdevinecop(n, obj, U = NULL, quasi = FALSE)

Arguments

Value

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

Author(s)

Thomas Nagler

References

Nagler, T., Czado, C. (2016)
Evading the curse of dimensionality in nonparametric density estimation.
Journal of Multivariate Analysis 151, 69-89 (doi:10.1016/j.jmva.2016.07.003)

Dissmann, J., Brechmann, E. C., Czado, C., and Kurowicka, D. (2013).
Selecting and estimating regular vine copulae and application to financial returns.
Computational Statistics & Data Analysis, 59(0):52–69.

See Also

kdevinecop, dkdecop, rkdecop, ghalton

Examples

data(wdbc, package = "kdecopula")                    # load data
u <- VineCopula::pobs(wdbc[, 5:7], ties = "average") # rank-transform

fit <- kdevinecop(u)                # estimate density
dkdevinecop(c(0.1, 0.1, 0.1), fit)  # evaluate density estimate

Univariate kernel density estimation for bounded and unbounded support

Description

Discrete variables are convoluted with the uniform distribution (see, Nagler, 2017). If a variable should be treated as discrete, declare it as ordered().

Usage

kde1d(x, mult = 1, xmin = -Inf, xmax = Inf, bw = NULL, bw_min = 0, ...)

Arguments

Details

If xmin or xmax are finite, the density estimate will be 0 outside of [xmin, xmax]. Mirror-reflection is used to correct for boundary bias. Discrete variables are convoluted with the uniform distribution (see, Nagler, 2017).

Value

An object of class kde1d.

References

Nagler, T. (2017). A generic approach to nonparametric function estimation with mixed data. arXiv:1704.07457

See Also

dkde1d, pkde1d, qkde1d, rkde1d plot.kde1d , lines.kde1d

Examples

data(wdbc, package = "kdecopula")  # load data
fit <- kde1d(wdbc[, 5])            # estimate density
dkde1d(1000, fit)                  # evaluate density estimate

Kernel density estimatior based on simplified vine copulas

Description

Implements the vine-copula based estimator of Nagler and Czado (2016). The marginal densities are estimated by kde1d, the vine copula density by kdevinecop. Discrete variables are convoluted with the uniform distribution (see, Nagler, 2017). If a variable should be treated as discrete, declare it as ordered(). Factors are expanded into binary dummy codes.

Usage

kdevine(x, mult_1d = NULL, xmin = NULL, xmax = NULL, copula.type = "kde", ...)

Arguments

Value

An object of class kdevine.

References

Nagler, T., Czado, C. (2016) Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas. Journal of Multivariate Analysis 151, 69-89 (doi:10.1016/j.jmva.2016.07.003)

Nagler, T. (2017). A generic approach to nonparametric function estimation with mixed data. arXiv:1704.07457

See Also

dkdevine kde1d kdevinecop

Examples

# load data
data(wdbc, package = "kdecopula")

# estimate density (use xmin to indicate positive support)
fit <- kdevine(wdbc[, 5:7], xmin = rep(0, 3))

# evaluate density estimate
dkdevine(c(1000, 0.1, 0.1), fit)

# plot simulated data
pairs(rkdevine(nrow(wdbc), fit))

Kernel estimation of vine copula densities

Description

The function estimates a vine copula density using kernel estimators for the pair copulas (based on the kdecopula package).

Usage

kdevinecop(
  data,
  matrix = NA,
  method = "TLL2",
  renorm.iter = 3L,
  mult = 1,
  test.level = NA,
  trunc.level = NA,
  treecrit = "tau",
  cores = 1,
  info = FALSE
)

Arguments

Value

An object of class kdevinecop. That is, a list containing

References

Nagler, T., Czado, C. (2016)
Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas.
Journal of Multivariate Analysis 151, 69-89 (doi:10.1016/j.jmva.2016.07.003)

Nagler, T., Schellhase, C. and Czado, C. (2017)
Nonparametric estimation of simplified vine copula models: comparison of methods arXiv:1701.00845

Dissmann, J., Brechmann, E. C., Czado, C., and Kurowicka, D. (2013).
Selecting and estimating regular vine copulae and application to financial returns.
Computational Statistics & Data Analysis, 59(0):52–69.

See Also

dkdevinecop, kdecop, BiCopIndTest, foreach

Examples

data(wdbc, package = "kdecopula")
# rank-transform to copula data (margins are uniform)
u <- VineCopula::pobs(wdbc[, 5:7], ties = "average")

fit <- kdevinecop(u)                   # estimate density
dkdevinecop(c(0.1, 0.1, 0.1), fit)     # evaluate density estimate
contour(fit)                           # contour matrix (Gaussian scale)
pairs(rkdevinecop(500, fit))           # plot simulated data

Plotting kde1d objects

Description

Plotting kde1d objects

Usage

## S3 method for class 'kde1d'
plot(x, ...)

## S3 method for class 'kde1d'
lines(x, ...)

Arguments

See Also

kde1d lines.kde1d

Examples

data(wdbc)  # load data
fit <- kde1d(wdbc[, 7])  # estimate density
plot(fit)  # plot density estimate

fit2 <- kde1d(as.ordered(wdbc[, 1])) # discrete variable
plot(fit2, col = 2)

Simulate from a kdevine object

Description

Simulate from a kdevine object

Usage

rkdevine(n, obj, quasi = FALSE)

Arguments

Value

An n x d matrix of simulated data from the kdevine object.

See Also

kdevine, rkdevinecop, rkde1d

Examples

# load and plot data
data(wdbc)

# estimate density
fit <- kdevine(wdbc[, 5:7], xmin = rep(0, 3))

# plot simulated data
pairs(rkdevine(nrow(wdbc), fit))

Wisconsin Diagnostic Breast Cancer (WDBC)

Description

The data contain measurements on cells in suspicious lumps in a women'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 classsified as either benign or malignant.

Usage

data(wdbc)

Format

wdbc is a data.frame with 31 columns. The first column indicates wether 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

https://archive.ics.uci.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)