Type: Package
Title: Kernel-Embedding-of-Probability Test for Elliptical Distribution
URL: https://github.com/tyy20/KEPTED
BugReports: https://github.com/tyy20/KEPTED/issues
Version: 0.2.0
Maintainer: Yin Tang <yqt5219@psu.edu>
Description: Provides an implementation of a kernel-embedding of probability test for elliptical distribution. This is an asymptotic test for elliptical distribution under general alternatives, and the location and shape parameters are assumed to be unknown. Some side-products are posted, including the transformation between rectangular and polar coordinates and two product-type kernel functions. See Tang and Li (2024) <doi:10.48550/arXiv.2306.10594> for details.
Encoding: UTF-8
License: BSD_3_clause + file LICENSE
Imports: expm, CompQuadForm, cubature, stats
RoxygenNote: 7.3.1
Suggests: knitr, rmarkdown, testthat (≥ 3.0.0)
Config/testthat/edition: 3
VignetteBuilder: knitr
NeedsCompilation: no
Packaged: 2024-04-12 02:10:39 UTC; yintang
Author: Yin Tang [aut, cre], Bing Li [cph, aut]
Repository: CRAN
Date/Publication: 2024-04-13 05:50:07 UTC

KEPTED: Kernel-Embedding-of-Probability Test for Elliptical Distribution

Description

Provides an implementation of a kernel-embedding of probability test for elliptical distribution. This is an asymptotic test for elliptical distribution under general alternatives, and the location and shape parameters are assumed to be unknown. Some side-products are posted, including the transformation between rectangular and polar coordinates and two product-type kernel functions. See Tang and Li (2024) doi:10.48550/arXiv.2306.10594 for details.

Author(s)

Maintainer: Yin Tang yqt5219@psu.edu

Authors:

See Also

Useful links:

Kernel embedding of probability test for elliptical distribution

Description

This function gives a test on whether the data is elliptically distributed based on kernel embedding of probability. See Tang and Li (2024) for details. Gaussian kernels and product-type inverse quadratic kernels are considered.

Usage

EllKEPT(
  X,
  eps = 1e-06,
  kerU = "Gaussian",
  kerTheta = "Gaussian",
  gamma.U = 0,
  gamma.Theta = 0
)

Arguments

X

A matrix with n rows and d columns.

eps

The regularization constant added to the diagonal to avoid singularity. Default value is 1e-6.

kerU

The type of kernel function on U. Currently supported options are "Gaussian" and "PIQ".

kerTheta

The type of kernel function on Theta. Currently supported options are "Gaussian" and "PIQ".

gamma.U

The tuning parameter gamma in the kernel function k_U(u1,u2). If gamma.U=0, the recommended procedure of selecting tuning parameter will be applied. Otherwise, the value given in gamma.U will be directly used as the tuning parameter. Default value is gamma.U=0. See "Details" for more information.

gamma.Theta

The tuning parameter gamma in the kernel function k_Theta(theta1,theta2). If gamma.Theta=0, the recommended procedure of selecting tuning parameter will be applied. Otherwise, the value given in gamma.Theta will be directly used as the tuning parameter. Default value is gamma.Theta=0. See "Details" for more information.

Details

The Gaussian kernel is defined as k(z1,z2)=exp(-gamma*||z1-z2||^2), and the Product-type Inverse-Quadratic (PIQ) kernel is defines as k(z1,z2)=Prod_j(1/(1+gamma*(z1_j-z2_j)^2)). The recommended procedure of selecting tuning parameter is given as in the simulation section of Tang and Li (2023+), where we set 1/sqrt(gamma)=(n(n-1)/2)^(-1)*sum_{1<=i<j<=n}||Z_i-Z_j||.

Value

A list of the following:

stat

The value of the test statistic.

pval

The p-value of the test.

lambda

The n eigenvalues in the approximated asymptotic distribution.

gamma.U

The tuning parameter gamma.U used in the test. Same as the input if its input is nonzero.

gamma.Theta

The tuning parameter gamma.Theta used in the test. Same as the input if its input is nonzero.

Note

In the arguments, eps refers to a regularization constant added to the diagonal. When the dimension is high, we recommend increasing eps to avoid singularity.

References

Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594

Examples

set.seed(313)
n=50
d=3

## Null Hypothesis
X=matrix(rnorm(n*d),nrow=n,ncol=d)
EllKEPT(X)

## Alternative Hypothesis
X=matrix(rchisq(n*d,2)-2,nrow=n,ncol=d)
EllKEPT(X)

Provides an implementation of a kernel-embedding of probability test for elliptical distribution. This is an asymptotic test for elliptical distribution under general alternatives, and the location and shape parameters are assumed to be unknown. Some side-products are posted, including the transformation between rectangular and polar coordinates and two product-type kernel functions.

Description

Provides an implementation of a kernel-embedding of probability test for elliptical distribution. This is an asymptotic test for elliptical distribution under general alternatives, and the location and shape parameters are assumed to be unknown. Some side-products are posted, including the transformation between rectangular and polar coordinates and two product-type kernel functions.

Author(s)

Yin Tang yqt5219@psu.edu Bing Li bxl9@psu.edu

References

Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594

Polar to rectangular coordinates

Description

Given a polar coordinate representation (R,Theta) of a d-dimensional vector X, where R is the length of X and the (d-1)-dimensional vector Theta contains the d-1 angles of X, this function compute X in its rectangular coordinate representation.

Usage

Polar2Rec(R, Theta)

Arguments

R

The length of X.

Theta

A vector of length d-1, containing the angles of X.

Details

The formula corresponds to v=rho(theta) as in Lemma 1 of Tang and Li (2024). See also Anderson (2003). Note that when d=2, V will be (sin(Theta),cos(Theta)).

Value

A list of the following:

X

A vector in rectangular coordinate.

V

The directional vector of X. Note that V is always on the unit sphere.

References

Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594 Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis. John Wiley & Suns, Inc. Huboken, New Jersey.

Examples

R=2
Theta=c(pi/6,pi/3)
Polar2Rec(R,Theta)

Derivative of the polar coordinate transformation

Description

This function compute the Jacobian matrix of the polar transformation theta=g(v), i.e., the transformation from the the rectangular coordinate representation of the directional vector to its angular representation.

Usage

PolarDerivative(v)

Arguments

v

A d-dimensional directional vector of length 1.

Details

See Lemma 3 of Tang and Li (2024).

Value

The Jacobian matrix of the polar transformation theta=g(v), with d-1 rows and d columns.

References

Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594

Examples

X=c(3,1,3)
V=X/sqrt(sum(X^2))
PolarDerivative(V)

Rectangular to polar coordinates

Description

Given a d-dimensional vector X in rectangular coordinate, this function compute its polar coordinate (R,Theta), where R is the length of X and the (d-1)-dimensional vector Theta contains the d-1 angles of X.

Usage

Rec2Polar(X)

Arguments

X

A vector in rectangular coordinate. Suppose the dimension of X is d.

Details

The formula corresponds to theta=g(v) as in Lemma 1 of Tang and Li (2024). See also Anderson (2003). Note that when d=2, V will be (sin(Theta),cos(Theta)).

Value

A list of the following:

R

The length of X.

Theta

A vector of length d-1, containing the angles of X.

References

Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594 Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis. John Wiley & Suns, Inc. Huboken, New Jersey.

Examples

X=c(3,1,3)
Rec2Polar(X)

Gaussian kernel computation

Description

Computing the values of Gaussian kernel functions.

Usage

kerGauss(gamma, z1, z2)

Arguments

gamma

A number, the bandwidth parameter in the Gaussian kernel.

z1

A vector, the first input of the Gaussian kernel.

z2

A vector, the second input of the Gaussian kernel.

Details

The Gaussian kernel is defined as k(z1,z2)=exp(-gamma*||z1-z2||^2).

Value

A number, the value of the Gaussian kernel function.

Examples

gamma=0.02
z1=c(3,1,3)
z2=c(8,1,9)
kerGauss(gamma,z1,z2)

Product-type Inverse-Quadratic (PIQ) kernel computation

Description

Computing the values of Product-type Inverse-Quadratic (PIQ) kernel functions.

Usage

kerPIQ(gamma, z1, z2)

Arguments

gamma

A number, the bandwidth parameter in the PIQ kernel.

z1

A vector, the first input of the PIQ kernel.

z2

A vector, the second input of the PIQ kernel.

Details

The Product-type Inverse-Quadratic (PIQ) kernel is defined as k(z1,z2)=Prod_j(1/(1+gamma*(z1_j-z2_j)^2)).

Value

A number, the value of the PIQ kernel function.

Examples

gamma=0.02
z1=c(3,1,3)
z2=c(8,1,9)
kerPIQ(gamma,z1,z2)