fitmssg {mixSSG}R Documentation

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

Description

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

Y=dμ+PλZ0+PΣ12Z1, {\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 μ\boldsymbol{\mu} (location vector in Rd{{{R}}}^{d}, λ\boldsymbol{\lambda} (skewness vector in Rd{{{R}}}^{d}), Σ\Sigma (positive definite symmetric dispersion matrix), and 0<α20<\alpha \leq 2 (tail thickness) are model parameters. Furthermore, PP is a positive stable random variable, Z0N(0,1){Z}_0\sim N({0},1), and Z1Nd(0,Σ)\bf{Z}_1\sim N_{d}\bigl({\bf{0}}, \Sigma\bigr). We note that ZZ, Z0Z_0, and Z1\boldsymbol{Z}_1 are mutually independent.

Usage

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

Arguments

Y

an n×dn\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 KK 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 KK clusters.

starts

a list of initial values if initial="TRUE". The list contains a vector of length KK of mixing (weight) parameters, a vector of length KK of tail thickness parameters, KK vectors of length of dd of location parameters, KK dispersion matrices, KK vectors of length of dd of skewness parameters, respectively.

Value

a list of estimated parameters corresponding to KK 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)


[Package mixSSG version 2.1.1 Index]