Title: Parsimonious Mixtures of MSEN and MTIN Distributions
Version: 1.0.0
Description: Implements parsimonious mixtures of MSEN and MTIN distributions via expectation- maximization based algorithms for model-based clustering. For each mixture component, parsimony is reached via the eigen-decomposition of the scale matrices and by imposing a constraint on the tailedness parameter. This produces a family of 28 parsimonious mixture models for each distribution.
License: GPL (≥ 3)
Encoding: UTF-8
LazyData: true
RoxygenNote: 7.1.1
Imports: doSNOW, foreach, snow, TSdist, tidyr, data.table, expint, zipfR, mclust, rlist, withr
Depends: R (≥ 2.10)
NeedsCompilation: no
Packaged: 2021-10-19 14:37:09 UTC; Daniele
Author: Salvatore D. Tomarchio [aut, cre], Bagnato Luca [aut], Antonio Punzo [aut]
Maintainer: Salvatore D. Tomarchio <daniele.tomarchio@unict.it>
Repository: CRAN
Date/Publication: 2021-10-20 06:50:02 UTC

Australian institute of sport data

Description

A dataset containing biometrical measurements for two categories of athletes collected at the Australian Institute of Sport.

Usage

data(AIS)

Format

A matrix with 202 observations on the following variables:

Sex

0 = Male or 1 = Female.

Ht

Height (in cm).

LBM

Lean body mass (in Kg).

RCC

Red cell count.

Hc

Hematocrit.

Hg

Hemoglobin.

SSF

Sum of skin folds.

Bfat

Body fat percentage.

Source

This dataset is a subset of the ais dataset contained in the alr4 R package.

References

Weisberg Sanford (2018). alr4: Data to Accompany Applied Linear Regression 4th Edition. https://CRAN.R-project.org/package=alr4.

Measurements on Two Hawk Species

Description

A dataset containing size-related measurements for two different Hawk species. Each species is further categorized by sex.

Usage

data(Hawks)

Format

A matrix with 323 observations on the following variables:

Class

1 = Male CH hawks, 2 = Male SS hawks, 3 = Female CH hawks or 4 = Female SS hawks

Wing

Length (in mm) of primary wing feather from tip to wrist it attaches to.

Weight

Body weight (in gm).

Tail

Measurement (in mm) related to the length of the tail.

Source

This dataset is a subset of the Hawks dataset contained in the Stat2Data R package.

References

Cannon et al. (2019). Stat2Data: Datasets for Stat2. https://CRAN.R-project.org/package=Stat2Data.

Fitting for parsimonious mixtures of MSEN or MTIN distributions

Description

Fits, by using EM-based algorithms, parsimonious mixtures of MSEN or MTIN distributions to the given data. Parallel computing is implemented and highly recommended for a faster model fitting. The Bayesian information criterion (BIC) and the integrated completed likelihood (ICL) are used to select the best fitting models according to each information criterion.

Usage

Mixt.fit(
  X,
  k = 1:3,
  init.par = NULL,
  cov.model = "all",
  theta.model = "all",
  density,
  ncores = 1,
  verbose = FALSE,
  ret.all = FALSE
)

Arguments

X

A data matrix with n rows and d columns, being n the number of data points and d the data the dimensionality.

k

An integer or a vector indicating the number of groups of the models to be estimated.

init.par

The initial values for starting the algorithms, as produced by the Mixt.fit.init() function.

cov.model

A character vector indicating the parsimonious structure of the scale matrices. Possible values are: "EII", "VII", "EEI", "VEI", "EVI", "VVI", "EEE", "VEE", "EVE", "EEV", "VVE", "VEV", "EVV", "VVV" or "all". When "all" is used, all of the 14 parsimonious structures are considered.

theta.model

A character vector indicating the parsimonious structure of the tailedness parameters. Possible values are: "E", "V" or "all". When "all" is used, both parsimonious structures are considered.

density

A character indicating the density of the mixture components. Possible values are: "MSEN" or "MTIN".

ncores

A positive integer indicating the number of cores used for running in parallel.

verbose

A logical indicating whether the running output should be displayed.

ret.all

A logical indicating whether to report the results of all the models or only those of the best models according to BIC and ICL.

Value

A list with the following elements:

all.models

The results related to the all the fitted models (only when ret.all=TRUE).

BicWin

The best fitting model according to the BIC.

IclWin

The best fitting model according to the ICL.

Summary

A quick table showing summary results for the best fitting models according to BIC and ICL.

Examples

set.seed(1234)
n <- 50
k <- 2
Pi <- c(0.5, 0.5)
mu <- matrix(c(0, 0, 4, 5), 2, 2)
cov.model <- "EEE"
lambda <- c(0.5, 0.5)
delta <- c(0.7, 0.7)
gamma <- c(2.62, 2.62)
theta <- c(0.1, 0.1)
density <- "MSEN"
data <- rMixt(n, k, Pi, mu, cov.model, lambda, delta, gamma, theta, density)

X <- data$X
nstartR <- 1
init.par <- Mixt.fit.init(X, k, density, nstartR)

theta.model <- "E"
res <- Mixt.fit(X, k, init.par, cov.model, theta.model, density)

Initialization for the EM-based algorithms

Description

Runs the initialization of the EM-based algorithms used for fitting parsimonious mixtures of MSEN or MTIN distributions. Parallel computing is implemented and highly recommended for a faster calculation.

Usage

Mixt.fit.init(X, k = 1:3, density, nstartR = 100, ncores = 1, verbose = FALSE)

Arguments

X

A data matrix with n rows and d columns, being n the number of data points and d the data the dimensionality.

k

An integer or a vector indicating the number of groups of the models.

density

A character indicating the density of the mixture components. Possible values are: "MSEN" or "MTIN".

nstartR

An integer specifying the number of random starts to be considered.

ncores

A positive integer indicating the number of cores used for running in parallel.

verbose

A logical indicating whether the running output should be displayed.

Value

init

A list of objects to be used by the Mixt.fit() function.

Examples

set.seed(1234)
n <- 50
k <- 2
Pi <- c(0.5, 0.5)
mu <- matrix(c(0, 0, 4, 5), 2, 2)
cov.model <- "EEE"
lambda <- c(0.5, 0.5)
delta <- c(0.7, 0.7)
gamma <- c(2.62, 2.62)
theta <- c(0.1, 0.1)
density <- "MSEN"
data <- rMixt(n, k, Pi, mu, cov.model, lambda, delta, gamma, theta, density)

X <- data$X
nstartR <- 1
init.par <- Mixt.fit.init(X, k, density, nstartR)

Density of a MSEN distribution

Description

Density of a MSEN distribution

Usage

dmsen(x, mu = rep(0, d), Sigma, theta = Inf, formula = "direct")

Arguments

x

A data matrix with n rows and d columns, being n the number of data points and d the data the dimensionality.

mu

A vector of length d representing the mean value.

Sigma

A symmetric positive-definite matrix representing the scale matrix of the distribution.

theta

A number greater than 0 indicating the tailedness parameter.

formula

Method used to calculate the density: "direct", "indirect", "series".

Value

The value(s) of the density in x

References

Punzo A., and Bagnato L. (2020). Allometric analysis using the multivariate shifted exponential normal distribution. Biometrical Journal, 62(6), 1525-1543.

Examples

d <- 3
x <- matrix(rnorm(d*2), 2, d)
dmsen(x, mu = rep(0,d), Sigma = diag(d), theta = 0.4, formula = "direct")

Density of a MTIN distribution

Description

Density of a MTIN distribution

Usage

dmtin(x, mu = rep(0, d), Sigma, theta = 0.01, formula = "direct")

Arguments

x

A data matrix with n rows and d columns, being n the number of data points and d the data the dimensionality.

mu

A vector of length d representing the mean value.

Sigma

A symmetric positive-definite matrix representing the scale matrix of the distribution.

theta

A number greater than 0 indicating the tailedness parameter.

formula

Method used to calculate the density: "direct", "indirect", "series".

Value

The value(s) of the density in x

References

Punzo A., and Bagnato L. (2021). The multivariate tail-inflated normal distribution and its application in finance. Journal of Statistical Computation and Simulation, 91(1), 1-36.

Examples

d <- 3
x <- matrix(rnorm(d*2), 2, d)
dmtin(x, mu = rep(0,d), Sigma = diag(d), theta = 0.9, formula = "direct")

Random number generation for bidimensional parsimonious mixtures of MSEN or MTIN distributions

Description

Random number generation for bidimensional parsimonious mixtures of MSEN or MTIN distributions

Usage

rMixt(n, k, Pi, mu, cov.model, lambda, delta, gamma, theta, density)

Arguments

n

An integer specifying the number of data points to be simulated.

k

An integer indicating the number of groups in the data.

Pi

A vector of length k representing the probability of belonging to the k groups for each data point.

mu

A matrix of means with 2 rows and k columns.

cov.model

A character indicating the parsimonious structure of the scale matrices. Possible values are: "EII", "VII", "EEI", "VEI", "EVI", "VVI", "EEE", "VEE", "EVE", "EEV", "VVE", "VEV", "EVV" or "VVV".

lambda

A numeric vector of length k, related to the scale matrices (see Punzo et al., 2016), which determines the volumes of the mixture components. Each element must be greater than 0. Required for all the parsimonious structures.

delta

A numeric vector of length k, related to the scale matrices (see Punzo et al., 2016), which determines the shapes of the mixture components. Each element must be between 0 and 1. Required for all the parsimonious structures, with the exclusion of "EII" and "VII".

gamma

A numeric vector of length k, related to the scale matrices (see Punzo et al., 2016), which determines the orientation of the mixture components. Each element represents an angle expressed in radian unit. Required for the "EEE", "VEE", "EVE", "EEV", "VVE", "VEV", "EVV" or "VVV" parsimonious structures.

theta

A vector of length k representing the tailedness parameters.

density

A character indicating the density of the mixture components. Possible values are: "MSEN" or "MTIN".

Value

A list with the following elements:

X

A data matrix with n rows and 2 columns.

Sigma

An array of dimension 2 x 2 x k for the generated scale matrices.

Size

The size of each generated group.

References

Punzo A., Browne R. and McNicholas P.D. (2016). Hypothesis Testing for Mixture Model Selection. Journal of Statistical Computation and Simulation, 86(14), 2797-2818.

Examples

n <- 50
k <- 2
Pi <- c(0.5, 0.5)
mu <- matrix(c(0, 0, 4, 5), 2, 2)
cov.model <- "EEE"
lambda <- c(0.5, 0.5)
delta <- c(0.7, 0.7)
gamma <- c(2.62, 2.62)
theta <- c(0.1, 0.1)
density <- "MSEN"
data <- rMixt(n, k, Pi, mu, cov.model, lambda, delta, gamma, theta, density)

Random number generation for the MSEN distribution

Description

Random number generation for the MSEN distribution

Usage

rmsen(n, mu = rep(0, d), Sigma, theta = Inf)

Arguments

n

An integer specifying the number of data points to be simulated.

mu

A vector of length d, where d is the dimensionality, representing the mean value.

Sigma

A symmetric positive-definite matrix representing the scale matrix of the distribution.

theta

A number greater than 0 indicating the tailedness parameter.

Value

A list with the following elements:

X

A data matrix with n rows and d columns.

w

A vector of weights of dimension n.

References

Punzo A., and Bagnato L. (2020). Allometric analysis using the multivariate shifted exponential normal distribution. Biometrical Journal, 62(6), 1525-1543.

Examples

d <- 3
rmsen(10, mu = rep(0, d), Sigma = diag(d), theta = 0.3)

Random number generation for the MTIN distribution

Description

Random number generation for the MTIN distribution

Usage

rmtin(n, mu = rep(0, d), Sigma, theta = 0.01)

Arguments

n

An integer specifying the number of data points to be simulated.

mu

A vector of length d, where d is the dimensionality, representing the mean value.

Sigma

A symmetric positive-definite matrix representing the scale matrix of the distribution.

theta

A number between 0 and 1 indicating the tailedness parameter.

Value

A list with the following elements:

X

A data matrix with n rows and d columns.

w

A vector of weights of dimension n.

References

Punzo A., and Bagnato L. (2021). The multivariate tail-inflated normal distribution and its application in finance. Journal of Statistical Computation and Simulation, 91(1), 1-36.

Examples

d <- 3
rmtin(10, mu = rep(0, d), Sigma = diag(d), theta = 0.9)

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