| Title: | 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. |
| Authors: | Yin Tang [aut, cre], Bing Li [cph, aut] |
| Maintainer: | Yin Tang <[email protected]> |
| License: | BSD_3_clause + file LICENSE |
| Version: | 0.2.0 |
| Built: | 2025-03-08 04:01:27 UTC |
| Source: | https://github.com/tyy20/kepted |
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.
EllKEPT( X, eps = 1e-06, kerU = "Gaussian", kerTheta = "Gaussian", gamma.U = 0, gamma.Theta = 0 )EllKEPT( X, eps = 1e-06, kerU = "Gaussian", kerTheta = "Gaussian", gamma.U = 0, gamma.Theta = 0 )
X |
A matrix with n rows and d columns. |
eps |
The regularization constant added to the diagonal to avoid
singularity. Default value is |
kerU |
The type of kernel function on |
kerTheta |
The type of kernel function on |
gamma.U |
The tuning parameter |
gamma.Theta |
The tuning parameter |
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||.
A list of the following:
stat |
The value of the test statistic. |
pval |
The p-value of the test. |
lambda |
The |
gamma.U |
The tuning parameter |
gamma.Theta |
The tuning parameter |
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.
Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594
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)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.
Yin Tang [email protected] Bing Li [email protected]
Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594
Computing the values of Gaussian kernel functions.
kerGauss(gamma, z1, z2)kerGauss(gamma, z1, z2)
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. |
The Gaussian kernel is defined as k(z1,z2)=exp(-gamma*||z1-z2||^2).
A number, the value of the Gaussian kernel function.
gamma=0.02 z1=c(3,1,3) z2=c(8,1,9) kerGauss(gamma,z1,z2)gamma=0.02 z1=c(3,1,3) z2=c(8,1,9) kerGauss(gamma,z1,z2)
Computing the values of Product-type Inverse-Quadratic (PIQ) kernel functions.
kerPIQ(gamma, z1, z2)kerPIQ(gamma, z1, z2)
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. |
The Product-type Inverse-Quadratic (PIQ) kernel is defined as k(z1,z2)=Prod_j(1/(1+gamma*(z1_j-z2_j)^2)).
A number, the value of the PIQ kernel function.
gamma=0.02 z1=c(3,1,3) z2=c(8,1,9) kerPIQ(gamma,z1,z2)gamma=0.02 z1=c(3,1,3) z2=c(8,1,9) kerPIQ(gamma,z1,z2)
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.
Polar2Rec(R, Theta)Polar2Rec(R, Theta)
R |
The length of |
Theta |
A vector of length |
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)).
A list of the following:
X |
A vector in rectangular coordinate. |
V |
The directional vector of |
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.
R=2 Theta=c(pi/6,pi/3) Polar2Rec(R,Theta)R=2 Theta=c(pi/6,pi/3) Polar2Rec(R,Theta)
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.
PolarDerivative(v)PolarDerivative(v)
v |
A |
See Lemma 3 of Tang and Li (2024).
The Jacobian matrix of the polar transformation theta=g(v), with
d-1 rows and d columns.
Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594
X=c(3,1,3) V=X/sqrt(sum(X^2)) PolarDerivative(V)X=c(3,1,3) V=X/sqrt(sum(X^2)) PolarDerivative(V)
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.
Rec2Polar(X)Rec2Polar(X)
X |
A vector in rectangular coordinate. Suppose the dimension of
|
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)).
A list of the following:
R |
The length of |
Theta |
A vector of length |
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.
X=c(3,1,3) Rec2Polar(X)X=c(3,1,3) Rec2Polar(X)