Title: Dirichlet-Based Polya Tree
Version: 1.0.1
Description: Contains functions to perform copula estimation by the non-parametric Bayesian method, Dirichlet-based Polya Tree. See Ning (2018) <doi:10.1080/00949655.2017.1421194>.
Depends: R (≥ 3.3.1)
License: MIT + file LICENSE
Encoding: UTF-8
LazyData: true
Imports: MCMCpack, stats, plyr, MASS, Rdpack
RdMacros: Rdpack
RoxygenNote: 6.0.1
NeedsCompilation: no
Packaged: 2018-06-19 02:29:13 UTC; ning
Author: Shaoyang Ning [aut, cre]
Maintainer: Shaoyang Ning <shaoyangning@fas.harvard.edu>
Repository: CRAN
Date/Publication: 2018-06-19 09:17:55 UTC

Calculating sub-partition probabiltiy measures for a realized distribution from D-P tree.

Description

DPTreeDensity returns the probablity measures in the finest sub-partitions of a realized distribution from D-P tree prior/posterior.

Usage

DPTreeDensity(Z)

Arguments

Z

An array of dimension of 2^m by 2^m by m, m being the approximation level. Realized Z's for all partitions at each level, as returned by RealizeDPTree.

Value

A 2^m by 2^m matrix. Normalized measures for all 2^m by 2^m sub-partititons on copula space given by the realized distribution from D-P tree.

References

Ning S and Shephard N (2018). “A nonparametric Bayesian approach to copula estimation.” Journal of Statistical Computation and Simulation, 88(6), pp. 1081-1105. doi: 10.1080/00949655.2017.1421194.

Examples

dp.rlz <- RealizeDPTree(DPTreePrior(m=2, z=1))
DPTreeDensity(dp.rlz)

Calculating sub-partition probabiltiy measures for the posterior mean distribution from D-P tree.

Description

DPTreePMeanDensity returns the probablity measures in the finest sub-partitions of the posterior mean from D-P tree.

Usage

DPTreePMeanDensity(prior)

Arguments

prior

A list. D-P tree specification. Should be in same format as returned from DPTreePrior or DPTreePosterior.

Value

A 2^m by 2^m matrix. Normalized measures for all 2^m by 2^m sub-partititons on copula space given by the posterior mean distribution from D-P tree.

References

Ning S and Shephard N (2018). “A nonparametric Bayesian approach to copula estimation.” Journal of Statistical Computation and Simulation, 88(6), pp. 1081-1105. doi: 10.1080/00949655.2017.1421194.

Examples

DPTreePMeanDensity(DPTreePrior(m=2, z=1))

D-P tree posterior updating from a single copula observation.

Description

DPTreePosterior returns the D-P tree posterior given input copula data.

Usage

DPTreePosterior(x, prior, w = 1)

Arguments

x

An array of length 2. Single copula data observation. Each element should be between 0 and 1.

prior

A list. Should be in same format as returned from DPTreePrior.

w

A positive number. Weight of data for posterior updating. Default 1.

Value

A list.

a

An array containing the hyperparameters of D-P trees.

References

Ning S and Shephard N (2018). “A nonparametric Bayesian approach to copula estimation.” Journal of Statistical Computation and Simulation, 88(6), pp. 1081-1105. doi: 10.1080/00949655.2017.1421194.

Examples

nsim = 1
rho = 0.9
data1 <- MASS::mvrnorm(n=nsim, mu=rep(0, 2), Sigma=matrix(c(1, rho, rho, 1), 2, 2))
data2 <- stats::pnorm(data1)
DPTreePosterior(x=data2, prior=DPTreePrior(m=4, z=1))

D-P tree posterior updating from multiple copula observations.

Description

DPTreePosteriorMulti returns the D-P tree posterior given input copula data.

Usage

DPTreePosteriorMulti(x, prior, w = 1)

Arguments

x

An array of dimension n by 2. Multiple copula data observations, with each row being a bivariate copula observation. All elements should be between 0 and 1.

prior

A list. Should be in same format as returned from DPTreePrior.

w

A positive number or an array of length n. Weight of data for posterior updating. Default 1.

Value

A list.

a

An array containing the hyperparameters of D-P trees.

References

Ning S and Shephard N (2018). “A nonparametric Bayesian approach to copula estimation.” Journal of Statistical Computation and Simulation, 88(6), pp. 1081-1105. doi: 10.1080/00949655.2017.1421194.

Examples

nsim = 10
rho = 0.9
data1 <- MASS::mvrnorm(n=nsim, mu=rep(0, 2), Sigma=matrix(c(1, rho, rho, 1), 2, 2))
data2 <- stats::pnorm(data1)
DPTreePosteriorMulti(x=data2, prior=DPTreePrior(m=4, z=1))

Generating the standard D-P Tree prior

Description

DPTreePrior returns a standard D-P Tree prior based on specified hyperparameters.

Usage

DPTreePrior(m = 4, z = 1)

Arguments

m

A positive integer. The finite approximation level for D-P tree. Default m=4.

z

A positive number. On i-th level, the hyperparameter for D-P tree prior is z\times i^2. Default z=1.

Value

A list.

a

An array containing the hyperparameters of D-P trees.

References

Ning S and Shephard N (2018). “A nonparametric Bayesian approach to copula estimation.” Journal of Statistical Computation and Simulation, 88(6), pp. 1081-1105. doi: 10.1080/00949655.2017.1421194.

Examples

DPTreePrior(m=6, z=1)

Sampling a realized distribution from the D-P Tree.

Description

RealizeDPTree returns a realized (copula) distribtuion sampled from the input D-P Tree.

Usage

RealizeDPTree(prior)

Arguments

prior

A list. Should be in same format as returned from DPTreePrior.

Value

An array of dimension 2^m by 2^m by m. m is the approximation level. Realized Z's for all partitions at each level. Three dimensions reprensent two marginals, and the level respectively.

References

Ning S and Shephard N (2018). “A nonparametric Bayesian approach to copula estimation.” Journal of Statistical Computation and Simulation, 88(6), pp. 1081-1105. doi: 10.1080/00949655.2017.1421194.

Examples

RealizeDPTree(DPTreePrior(m=2, z=1))

Sample a copula observation from a realized distribution from D-P tree.

Description

SampleDPTreeDensity returns a copula sample from a realized distribution from D-P tree.

Usage

SampleDPTreeDensity(nsam, d)

Arguments

nsam

A positive integer. The sample size.

d

A 2^m by 2^m matrix, m being the approximating level. Normalized measures for all 2^m by 2^m sub-partititons on copula space given by the realized distribution from D-P tree, as returned by DPTreeDensity.

Value

An array of dimension nsam by 2. The values of PDF of the input D-P tree distribution evaluated at the input points.

References

Ning S and Shephard N (2018). “A nonparametric Bayesian approach to copula estimation.” Journal of Statistical Computation and Simulation, 88(6), pp. 1081-1105. doi: 10.1080/00949655.2017.1421194.

Examples

SampleDPTreeDensity(10, DPTreePMeanDensity(DPTreePrior(m=2, z=1)))

The disitribution function for realized distribution from D-P tree.

Description

dDPTreeRealize returns the value of density function of realized distribution from D-P tree at certain given point on copula space.

Usage

dDPTreeRealize(d, x)

Arguments

d

A 2^m by 2^m matrix, m being the approximating level. Normalized measures for all 2^m by 2^m sub-partititons on copula space given by the realized distribution from D-P tree, as returned by DPTreeDensity.

x

An array of dimension n by 2. The points on copula space for density function evluation. Should be between 0 and 1.

Value

An array of length n. The values of PDF of the input D-P tree distribution evaluated at the input points.

References

Ning S and Shephard N (2018). “A nonparametric Bayesian approach to copula estimation.” Journal of Statistical Computation and Simulation, 88(6), pp. 1081-1105. doi: 10.1080/00949655.2017.1421194.

Examples

dDPTreeRealize(DPTreePMeanDensity(DPTreePrior(m=2, z=1)),c(0.5,0.5))

The disitribution function for realized distribution from D-P tree.

Description

pDPTreeRealize returns the value of distribution function of realized distribution from D-P tree at certain given point on copula space.

Usage

pDPTreeRealize(d, x)

Arguments

d

A 2^m by 2^m matrix, m being the approximating level. Normalized measures for all 2^m by 2^m sub-partititons on copula space given by the realized distribution from D-P tree, as returned by DPTreeDensity.

x

An array of dimension n by 2. The points on copula space for distribution function evluation. Should be between 0 and 1.

Value

An array of length n. The values of CDF of the input D-P tree distribution evaluated at the input points.

References

Ning S and Shephard N (2018). “A nonparametric Bayesian approach to copula estimation.” Journal of Statistical Computation and Simulation, 88(6), pp. 1081-1105. doi: 10.1080/00949655.2017.1421194.

Examples

pDPTreeRealize(DPTreePMeanDensity(DPTreePrior(m=2, z=1)),c(0.5,0.5))