Package 'mixSSG'

AIS data

Description

The set of AIS data involves recorded body factors of 202 athletes including 100 women 102 men, see Cook (2009). Among factors, two variables body mass index (BMI) and body fat percentage (Bfat) are chosen for cluster analysis.

Usage

data(AIS)

Format

A text file with 3 columns.

References

R. D. Cook and S. Weisberg, (2009). An Introduction to Regression Graphics, John Wiley & Sons, New York.

Examples

data(AIS)

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bankruptcy data

Description

The bankruptcy dataset involves ratio of the retained earnings (RE) to the total assets, and the ratio of earnings before interests and the taxes (EBIT) to the total assets of 66 American firms, see Altman (1969).

Usage

data(bankruptcy)

Format

A text file with 3 columns.

References

E. I. Altman, 1969. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy, The Journal of Finance, 23(4), 589-609.

Examples

data(bankruptcy)

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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 $f_{\boldsymbol{Y}}(\boldsymbol{y} | \boldsymbol{\Theta})$ for ${\boldsymbol{\Theta}}=(\alpha,\boldsymbol{\mu},\Sigma, \boldsymbol{\lambda})$ where $\alpha$, $\boldsymbol{\mu}$, $\Sigma$, and $\boldsymbol{\lambda}$ are tail thickness, location, dispersion matrix, and skewness parameters, respectively. Herein, we give a good approximation for $f_{\boldsymbol{Y}}(\boldsymbol{y} | \boldsymbol{\Theta})$. First , for ${\cal{N}}=50$, define

$L=\frac{\Gamma(\frac{{\cal{N}}\alpha}{2}+1+\frac{\alpha}{2})\Gamma\bigl(\frac{d+{\cal{N}}\alpha}{2}+\frac{\alpha}{2}\bigr)}{ \Gamma(\frac{{\cal{N}}\alpha}{2}+1)\Gamma\bigl(\frac{d+{\cal{N}}\alpha}{2}\bigr)({\cal{N}}+1)}.$

If $d(\boldsymbol{y})\leq 2L^{\frac{2}{\alpha}}$, then

$f_{\boldsymbol{Y}}(\boldsymbol{y} | {\boldsymbol{\Theta}}) \simeq \frac{{C}_{0}\sqrt{2\pi \delta }}{N} \sum_{i=1}^{N} \exp\Bigl\{-\frac{d(\boldsymbol{y})}{2p_{i}}\Bigr\}\Phi \bigl( m| 0, \sqrt{\delta p_{i}} \bigr)p_{i}^{-\frac{d}{2}},$

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 $d(\boldsymbol{y})> 2L^{\frac{2}{\alpha}}$, we have

$f_{\boldsymbol{Y}}(\boldsymbol{y} | \boldsymbol{\Theta})\simeq \frac{{C}_{0}\sqrt{d(\boldsymbol{y})\delta}}{\sqrt{\pi}} \sum_{j=1}^{{\cal{N}}}\frac{ (-1)^{j-1}\Gamma(\frac{j\alpha}{2}+1)\sin \bigl(\frac{j\pi \alpha}{2}\bigr)} {\Gamma(j+1)\bigl[\frac{d(\boldsymbol{y})}{2}\bigr]^{\frac{d+1+j\alpha}{2}}}\Gamma\Bigl(\frac{d+j\alpha}{2}\Bigr) T_{d+j\alpha}\biggl(m\sqrt{\frac{d+j\alpha}{d(\boldsymbol{y})\delta}}\biggr),$

where $T_{\nu}(x)$ is distribution function of the Student's $t$ with $\nu$ degrees of freedom, $\Phi(x|a,b)$ is the cumulative density function of normal distribution wih mean $a$ and standard deviation $b$, and ${C_{0}=2 (2\pi)^{-\frac{d+1}{2}}|{\Sigma}|^{-\frac{1}{2}},}$ $d(\boldsymbol{y})=(\boldsymbol{y}-\boldsymbol{\mu})^{'}{{\Omega}^{-1}}(\boldsymbol{y}-\boldsymbol{\mu}),$ ${m}=\boldsymbol{\lambda}^{'}{{\Omega}}^{-1}(\boldsymbol{y}-\boldsymbol{\mu}),$ ${\Omega}={\Sigma}+\boldsymbol{\lambda}\boldsymbol{\lambda}^{'},$ ${\delta}=1-\boldsymbol{\lambda}^{'}{\Omega}^{-1}\boldsymbol{\lambda}$.

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 $\alpha$-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)

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Estimating parameters of the symmetric $\alpha$-stable (S$\alpha$S) distribution using Bayesian paradigm.

Description

Let ${{y}}_1,{{y}}_2, \cdots,{{y}}_n$ are $n$ realizations form S$\alpha$S distribution with parameters $\alpha, \sigma$, and $\mu$. Herein, we estimate parameters of symmetric univariate stable distribution within a Bayesian framework. We consider a uniform distribution for prior of tail thickness, that is $\alpha \sim U(0,2)$. The normal and inverse gamma conjugate priors are designated for $\mu$ and $\sigma^2$ with density functions given, respectively, by

$\pi(\mu)=\frac{1}{\sqrt{2\pi}\sigma_{0}}\exp\Bigl\{-\frac{1}{2}\Bigl(\frac{\mu-\mu_0}{\sigma_0}\Bigr)^{2}\Bigr\},$

and

$\pi(\delta)= \delta_{0}^{\gamma_{0}}\delta^{-\gamma_0-1}\exp\Bigl\{-\frac{\delta_0}{\delta}\Bigr\},$

where $\mu_0 \in R$, $\sigma_0>0$, $\delta=\sigma^2$, $\delta_0>0$, and $\gamma_0>0$.

Usage

fitBayes(y, mu0, sigma0, gamma0, delta0, epsilon)

Arguments

y	vector of realizations that following S$\alpha$S distribution.
mu0	the location hyperparameter corresponding to $\pi(\mu)$.
sigma0	the standard deviation hyperparameter corresponding to $\pi(\mu)$.
gamma0	the shape hyperparameter corresponding to $\pi(\delta)$.
delta0	the rate hyperparameter corresponding to $\pi(\delta)$.
epsilon	a positive small constant playing the role of threshold for stopping sampler.

Value

Estimated tail thickness, location, and scale parameters, number of iterations to attain convergence, the log-likelihood value across iterations, the Bayesian information criterion (BIC), and the Akaike information criterion (AIC).

Author(s)

Mahdi Teimouri

Examples

n <- 100
alpha <- 1.4
mu <- 0
sigma <- 1
y <- rnorm(n)
fitBayes(y, mu0 = 0, sigma0 = 0.2, gamma0 = 10e-5, delta0 = 10e-5, epsilon = 0.005)

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Computing the maximum likelihood estimator for the mixtures of skewed sub-Gaussian stable distributions using the EM algorithm.

Description

Each $d$-dimensional skewed sub-Gaussian stable (SSG) random vector $\bf{Y}$, admits the representation given by Teimouri (2022):

${\bf{Y}} \mathop=\limits^d {\boldsymbol{\mu}}+\sqrt{P}{\boldsymbol{\lambda}}\vert{Z}_0\vert + \sqrt{P}{\Sigma}^{\frac{1}{2}}{\bf{Z}}_1,$

where $\boldsymbol{\mu}$ (location vector in ${{{R}}}^{d}$, $\boldsymbol{\lambda}$ (skewness vector in ${{{R}}}^{d}$), $\Sigma$ (positive definite symmetric dispersion matrix), and $0<\alpha \leq 2$ (tail thickness) are model parameters. Furthermore, $P$ is a positive stable random variable, ${Z}_0\sim N({0},1)$, and $\bf{Z}_1\sim N_{d}\bigl({\bf{0}}, \Sigma\bigr)$. We note that $Z$, $Z_0$, and $\boldsymbol{Z}_1$ are mutually independent.

Usage

fitmssg(Y, K, eps = 0.15, initial = "FALSE", method = "moment", starts = starts)

Arguments

Y	an $n\times d$ matrix of observations.
K	number of component.
eps	threshold value for stopping EM algorithm. It is 0.15 by default. The algorithm can be implemented faster if eps is larger.
initial	logical statement. If initial = TRUE, then a list of the initial values must be given. Otherwise, it is determined by method.
method	either em or moment. If method = "moment", then the initial values are determined through the method of moment applied to each of $K$ clusters that are obtained through the k-means method of Hartigan and Wong (1979). Otherwise, the initial values for each cluster are determined through the EM algorithm (Teimouri et al., 2018) developed for sub-Gaussian stable distributions applied to each of $K$ clusters.
starts	a list of initial values if initial="TRUE". The list contains a vector of length $K$ of mixing (weight) parameters, a vector of length $K$ of tail thickness parameters, $K$ vectors of length of $d$ of location parameters, $K$ dispersion matrices, $K$ vectors of length of $d$ of skewness parameters, respectively.

Value

a list of estimated parameters corresponding to $K$ clusters, predicted labels for clusters, the log-likelihood value across iterations, the Bayesian information criterion (BIC), and the Akaike information criterion (AIC).

Author(s)

Mahdi Teimouri

References

M. Teimouri, 2022. Finite mixture of skewed sub-Gaussian stable distributions, arxiv.org/abs/2205.14067.

M. Teimouri, S. Rezakhah, and A. Mohammadpour, 2018. Parameter estimation using the EM algorithm for symmetric stable random variables and sub-Gaussian random vectors, Journal of Statistical Theory and Applications, 17(3), 439-41.

J. A. Hartigan, M. A. Wong, 1979. Algorithm as 136: A k-means clustering algorithm, Journal of the Royal Statistical Society. Series c (Applied Statistics), 28, 100-108.

Examples

data(bankruptcy)
out1<-fitmssg(bankruptcy[,2:3], K=2, eps = 0.15, initial="FALSE", method="moment", starts=starts)
n1 <- 100
n2 <- 50
omega1 <- n1/(n1 + n2)
omega2 <- n2/(n1 + n2)
alpha1 <- 1.6
alpha2 <- 1.6
mu1 <- c(-1, -1)
mu2 <- c(6, 6)
sigma1 <- matrix( c(2, 0.20, 0.20, 0.5), 2, 2 )
sigma2 <- matrix( c(0.4, 0.10, 0.10, 0.2  ), 2, 2 )
lambda1 <- c(5, 5)
lambda2 <- c(-5, -5)
Sigma <- array( NA, c(2, 2, 2) )
Sigma[, , 1] <- sigma1
Sigma[, , 2] <- sigma2
starts<-list( c(omega1,omega2), c(alpha1,alpha2), rbind(mu1,mu2), Sigma, rbind(lambda1,lambda2) )
Y <- rbind( rssg(n1 , alpha1, mu1, sigma1, lambda1),  rssg(n2, alpha2, mu2, sigma2, lambda2) )
out2<-fitmssg(Y, K=2, eps=0.15, initial="TRUE", method="moment", starts=starts)

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Simulating positive stable random variable.

Description

The cumulative distribution function of positive stable distribution is given by

$F_{P}(x)=\frac{1}{\pi}\int_{0}^{\pi}\exp\Bigl\{-x^{-\frac{\alpha}{2-\alpha}}a(\theta)\Bigr\}d\theta,$

where $0<\alpha \leq 2$ is tail thickness or index of stability and

$a(\theta)=\frac{\sin\Bigl(\bigl(1-\frac{\alpha}{2}\bigr)\theta\Bigr)\Bigl[\sin \bigl(\frac{\alpha \theta}{2}\bigr)\Bigr]^{\frac{\alpha}{2-\alpha}}}{[\sin(\theta)]^{\frac{2}{2-\alpha}}}.$

Kanter (1975) used the above integral transform to simulate positive stable random variable as

$P\mathop=\limits^d\Bigl( \frac{a(\theta)}{W} \Bigr)^{\frac{2-\alpha}{\alpha}},$

in which $\theta\sim U(0,\pi)$ and $W$ independently follows an exponential distribution with mean unity.

Usage

rpstable(n, alpha)

Arguments

n	the number of samples required.
alpha	the tail thickness parameter.

Value

simulated realizations of size $n$ from positive $\alpha$-stable distribution.

Author(s)

Mahdi Teimouri

References

M. Kanter, 1975. Stable densities under change of scale and total variation inequalities, Annals of Probability, 3(4), 697-707.

Examples

rpstable(10, alpha = 1.2)

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Simulating skewed sub-Gaussian stable random vector.

Description

Each skewed sub-Gaussian stable (SSG) random vector $\bf{Y}$, admits the representation

${\bf{Y}} \mathop=\limits^d {\boldsymbol{\mu}}+\sqrt{P}{\boldsymbol{\lambda}}\vert{Z}_0\vert + \sqrt{P}{\Sigma}^{\frac{1}{2}}{\bf{Z}}_1,$

where ${\boldsymbol{\mu}} \in {R}^{d}$ is location vector, ${\boldsymbol{\lambda}} \in {R}^{d}$ is skewness vector, $\Sigma$ is a positive definite symmetric dispersion matrix, and $0<\alpha \leq 2$ is tail thickness. Further, $P$ is a positive stable random variable, ${Z}_0\sim N({0},1)$, and ${\bf{Z}}_1\sim N_{d}\bigl({\bf{0}}, \Sigma\bigr)$. We note that $Z$, $Z_0$, and ${\bf{Z}}_1$ are mutually independent.

Usage

rssg(n, alpha, Mu, Sigma, Lambda)

Arguments

n	the number of samples required.
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 the skewed sub-Gaussian stable distribution.

Author(s)

Mahdi Teimouri

Examples

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

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Estimating the tail index of the skewed sub-Gaussian stable distribution using the stochastic EM algorithm given that other parameters are known.

Description

Suppose ${\boldsymbol{Y}}_1,{\boldsymbol{Y}}_2, \cdots,{\boldsymbol{Y}}_n$ are realizations following $d$-dimensional skewed sub-Gaussian stable distribution. Herein, we estimate the tail thickness parameter $0<\alpha \leq 2$ when $\boldsymbol{\mu}$ (location vector in ${{{R}}}^{d}$, $\boldsymbol{\lambda}$ (skewness vector in ${{{R}}}^{d}$), and $\Sigma$ (positive definite symmetric dispersion matrix are assumed to be known.

Usage

stoch(Y, alpha0, Mu0, Sigma0, Lambda0)

Arguments

Y	a vector (or an $n\times d$ matrix) at which the density function is approximated.
alpha0	initial value for the tail thickness parameter.
Mu0	a vector giving the initial value for the location parameter.
Sigma0	a positive definite symmetric matrix specifying the initial value for the dispersion matrix.
Lambda0	a vector giving the initial value for the skewness parameter.

Details

Here, we assume that parameters ${\boldsymbol{\mu}}$, ${\boldsymbol{\lambda}}$, and $\Sigma$ are known and only the tail thickness parameter needs to be estimated.

Value

Estimated tail thickness parameter $\alpha$, of the skewed sub-Gaussian stable distribution.

Author(s)

Mahdi Teimouri

Examples

n <- 100
alpha <- 1.4
Mu <- rep(0, 2)
Sigma <- diag(2)
Lambda <- rep(2, 2)
Y <- rssg(n, alpha, Mu, Sigma, Lambda)
stoch(Y, alpha, Mu, Sigma, Lambda)

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