dssg: Approximating the density function of skewed sub-Gaussian...

In mixSSG: Clustering Using Mixtures of Sub Gaussian Stable Distributions

Approximating the density function of skewed sub-Gaussian stable distribution.

Description

Suppose d-dimensional random vector \boldsymbol{Y} follows a skewed sub-Gaussian stable distribution with density function f_{\boldsymbol{Y}}(\boldsymbol{y} | \boldsymbol{Θ}) for {\boldsymbol{Θ}}=(α,\boldsymbol{μ},Σ, \boldsymbol{λ}) where α, \boldsymbol{μ}, Σ, and \boldsymbol{λ} are tail thickness, location, dispersion matrix, and skewness parameters, respectively. Herein, we give a good approximation for f_{\boldsymbol{Y}}(\boldsymbol{y} | \boldsymbol{Θ}). First , for {\cal{N}}=50, define

L=\frac{Γ(\frac{{\cal{N}}α}{2}+1+\frac{α}{2})Γ\bigl(\frac{d+{\cal{N}}α}{2}+\frac{α}{2}\bigr)}{ Γ(\frac{{\cal{N}}α}{2}+1)Γ\bigl(\frac{d+{\cal{N}}α}{2}\bigr)({\cal{N}}+1)}.

If d(\boldsymbol{y})≤q 2L^{\frac{2}{α}}, then

f_{\boldsymbol{Y}}(\boldsymbol{y} | {\boldsymbol{Θ}}) \simeq \frac{{C}_{0}√{2π δ }}{N} ∑_{i=1}^{N} \exp\Bigl\{-\frac{d(\boldsymbol{y})}{2p_{i}}\Bigr\}Φ \bigl( m| 0, √{δ p_{i}} \bigr)p_{i}^{-\frac{d}{2}},

where, p_1,p_2,\cdots, p_N (for N=3000) are independent realizations following positive stable distribution that are generated using command rpstable(3000, alpha). Otherwise, if d(\boldsymbol{y})> 2L^{\frac{2}{α}}, we have

f_{\boldsymbol{Y}}(\boldsymbol{y} | \boldsymbol{Θ})\simeq \frac{{C}_{0}√{d(\boldsymbol{y})δ}}{√{π}} ∑_{j=1}^{{\cal{N}}}\frac{ (-1)^{j-1}Γ(\frac{jα}{2}+1)\sin \bigl(\frac{jπ α}{2}\bigr)} {Γ(j+1)\bigl[\frac{d(\boldsymbol{y})}{2}\bigr]^{\frac{d+1+jα}{2}}}Γ\Bigl(\frac{d+jα}{2}\Bigr) T_{d+jα}\biggl(m√{\frac{d+jα}{d(\boldsymbol{y})δ}}\biggr),

where T_{ν}(x) is distribution function of the Student's t with ν degrees of freedom, Φ(x|a,b) is the cumulative density function of normal distribution wih mean a and standard deviation b, and {C_{0}=2 (2π)^{-\frac{d+1}{2}}|{Σ}|^{-\frac{1}{2}},} d(\boldsymbol{y})=(\boldsymbol{y}-\boldsymbol{μ})^{'}{{Ω}^{-1}}(\boldsymbol{y}-\boldsymbol{μ}), {m}=\boldsymbol{λ}^{'}{{Ω}}^{-1}(\boldsymbol{y}-\boldsymbol{μ}), {Ω}={Σ}+\boldsymbol{λ}\boldsymbol{λ}^{'}, {δ}=1-\boldsymbol{λ}^{'}{Ω}^{-1}\boldsymbol{λ}.

Usage

dssg(Y, alpha, Mu, Sigma, Lambda)

Arguments

Y	a vector (or an n\times d matrix) at which the density function is approximated.
alpha	the tail thickness parameter.
Mu	a vector giving the location parameter.
Sigma	a positive definite symmetric matrix specifying the dispersion matrix.
Lambda	a vector giving the skewness parameter.

Value

simulated realizations of size n from positive α-stable distribution.

Author(s)

Mahdi Teimouri

Examples

n <- 4
alpha <- 1.4
Mu <- rep(0, 2)
Sigma <- diag(2)
Lambda <- rep(2, 2)
Y <- rssg(n, alpha, Mu, Sigma, Lambda)
dssg(Y, alpha, Mu, Sigma, Lambda)

mixSSG documentation built on Sept. 11, 2022, 5:06 p.m.