Type:	Package
Title:	Constrained Mixture of Generalized Normal Distributions
Version:	0.1.1
Description:	The 'cmgnd' implements the constrained mixture of generalized normal distributions model, a flexible statistical framework for modelling univariate data exhibiting non-normal features such as skewness, multi-modality, and heavy tails. By imposing constraints on model parameters, the 'cmgnd' reduces estimation complexity while maintaining high descriptive power, offering an efficient solution in the presence of distributional irregularities. For more details see Duttilo and Gattone (2025) <doi:10.1007/s00180-025-01638-x> and Duttilo et al (2025) <doi:10.48550/arXiv.2506.03285>.
Depends:	R (≥ 4.0.0)
Imports:	RcppAlgos, lubridate, stats, utils, gnorm, purrr, ggplot2
Encoding:	UTF-8
RoxygenNote:	7.2.3
License:	GPL (≥ 3)
URL:	https://github.com/pierdutt/cmgnd
BugReports:	https://github.com/pierdutt/cmgnd/issues
NeedsCompilation:	no
Packaged:	2025-07-15 09:57:33 UTC; User
Author:	Pierdomenico Duttilo [aut, cre], Stefano Antonio Gattone [aut], Alfred Kume [aut]
Maintainer:	Pierdomenico Duttilo <pierdomenico.duttilo@unipd.it>
Repository:	CRAN
Date/Publication:	2025-07-18 14:50:02 UTC

cmgnd: Function for Clustering using Constrained Mixtures of Generalized Normal Distributions

Description

Fits univariate constrained mixture of generalized normal distribution models by imposing mixture partitions. Models are estimated by the ECM algorithm initialized by k-means.

Usage

cmgnd(
  x,
  K = 2,
  Cmu = rep(0, K),
  Csigma = rep(0, K),
  Cnu = rep(0, K),
  nstart = 50,
  theta = FALSE,
  nustart = rep(2, K),
  nustartype = "random",
  gauss = FALSE,
  laplace = FALSE,
  scale = FALSE,
  eps = 10^-4,
  maxit = 999,
  verbose = TRUE,
  sigbound = c(0.1, 5),
  sr = "like",
  eta = 0.5,
  seed = 12345,
  seed.nstart = seq(1:nstart)
)

Arguments

Details

The constrained mixture of generalized normal distributions (CMGND) model is an advanced statistical tool designed for analyzing univariate data characterized by non-normal features such as asymmetry, multi-modality, leptokurtosis, and heavy tails. This model extends the mixture of generalized normal distributions (MGND) by incorporating constraints on the parameters, thereby reducing the number of parameters to be estimated and improving model performance. The CMGND model is defined by the following components:

f(x|\theta) = \sum_{k=1}^{K} \pi_k f_k(x|\mu_k, \sigma_k, \nu_k)

where: \pi_k are the mixture weights, satisfying 0 < \pi_k < 1 and \sum_{k=1}^{K} \pi_k = 1. f_k(x|\mu_k, \sigma_k, \nu_k) is the Generalized Normal Distribution for the k-th component with mean \mu_k, scale \sigma_k, and shape parameter \nu_k.

The parameter space can be constrained by imposing equality constraints such as \mu_k = \mu_r, \sigma_k = \sigma_r, and/or \nu_k = \nu_r for all k \in C_r, where C_r is a partition of the set \{1, 2, \ldots, K\}.

The k \in C_r partition for each parameter can be specified by the binary vectors Cmu, Csigma and Cnu.

Value

References

Bazi, Y., Bruzzone, L., and Melgani, F. (2006). Image thresholding based on the em algorithm and the generalized gaussian distribution. Pattern Recognition, 40(2), pp 619–634.

Wen, L., Qiu, Y., Wang, M., Yin, J., and Chen, P. (2022). Numerical characteristics and parameter estimation of finite mixed generalized normal distribution. Communications in Statistics - Simulation and Computation, 51(7), pp 3596–3620.

Duttilo, P. (2024). Modelling financial returns with mixtures of generalized normal distributions. PhD Thesis, University “G. d’Annunzio” of Chieti-Pescara, pp. 1-166, doi:10.48550/arXiv.2411.11847

Duttilo, P. and Gattone, S.A. (2025). Enhancing parameter estimation in finite mixture of generalized normal distributions, Computational Statistics, pp. 1-28, doi:10.1007/s00180-025-01638-x

Duttilo, P., Gattone, S.A., and Kume A. (2025). Constrained mixtures of generalized normal distributions, pp. 1-34, doi:10.48550/arXiv.2506.03285

Examples

# Old Faithful dataset
x=faithful$eruptions
# Unconstrained model estimation
Cmu <- c(0, 0)
Csigma <- c(0, 0)
Cnu <- c(0, 0)
model_unc <- cmgnd(x, nstart = 2, K = 2, Cmu, Csigma, Cnu)
model_unc$parameters
plot_cmgnd(x, model_unc)
# Constrained model estimation with common scale parameters
Csigma <- c(1, 1)
model_con <- cmgnd(x, nstart = 2, K =2, Cmu, Csigma, Cnu)
model_con$parameters
plot_cmgnd(x, model_con)

Marginal Density Estimation for CMGND Models

Description

This function estimates the marginal density for univariate constrained mixture of generalized normal distribution (CMGND) models.

Usage

dcmgnd(x, parameters)

Arguments

Details

The function computes the marginal density based on the provided parameters of the CMGND model. It can handle both newly supplied parameters or those extracted from an existing CMGND model object.

Value

A vector of density estimates corresponding to the input data 'x'.

See Also

'cmgnd()' for estimating the model parameters.

The Generalized Normal Distribution (GND)

Description

Density function for the GND with location parameter mu, scale parameter sigma and shape parameter nu.

Usage

dgnd(x, mu = 0, sigma = 1, nu = 2)

Arguments

Details

If mu, sigma and nu are not specified they assume the default values of 0, 1 and 2, respectively. The GND distribution has density

f_{GND}(x|\mu,\sigma,\nu)=\frac{\nu}{2\sigma\Gamma(1\mathbin{/}\nu)}\exp\Biggr\{-\Biggr|\frac{x-\mu}{\sigma}\Biggr|^\nu\Biggr\}.

The shape parameter \nu controls both the peakedness and tail weights. If \nu=1 the GND reduces to the Laplace distribution and if \nu=2 it coincides with the normal distribution. It is noticed that 1<\nu<2 yields an intermediate distribution between the normal and the Laplace distribution. As limit cases, for \nu\rightarrow\infty the distribution tends to a uniform distribution, while for \nu\rightarrow0 it will be impulsive.

Value

dgnd returns the density.

References

Nadarajah, S. (2005). A generalized normal distribution. Journal of Applied Statistics, 32(7):685–694.

Plot Marginal and Mixture Component Densities of the CMGND Model

Description

This function generates a plot displaying both the marginal density and individual mixture component densities for univariate constrained mixture of generalized normal distribution (CMGND) models. It visually represents how the different components of the mixture model contribute to the overall density.

Usage

hist_cmgnd(x, parameters, bins = 80)

Arguments

Details

The function plots the overall (marginal) density curve for the CMGND model, as well as the density curves of each mixture component. This visualization helps in understanding how each component contributes to the model and provides insights into the data distribution.

Value

A plot illustrating the marginal density along with the densities of the individual mixture components for the given data 'x'.

See Also

'cmgnd()' for estimating the model parameters.

Compute the First Four Moments of the CMGND Marginal Distribution

Description

Computes the mean, variance, skewness, and kurtosis of the marginal distribution of a univariate Constrained Mixture of Generalized Normal Distributions (CMGND) model, given the model parameters.

Usage

moments_cmgnd(parameters)

Arguments

Details

The function assumes that the parameters define a valid CMGND model and uses analytical expressions to compute the first four moments of the marginal distribution. The shape parameter \nu_k governs the kurtosis of each component.

Value

A named list with the following elements:

mean

The marginal mean of the CMGND distribution

var

The marginal variance

skew

The marginal skewness

kur

The marginal kurtosis

See Also

cmgnd for estimating CMGND model parameters.

Plot Marginal and Mixture Component Densities of the CMGND Model

Description

This function generates a plot displaying both the marginal density and individual mixture component densities for univariate constrained mixture of generalized normal distribution (CMGND) models. It visually represents how the different components of the mixture model contribute to the overall density.

Usage

plot_cmgnd(x, parameters, model = "")

Arguments

Details

The function plots the overall (marginal) density curve for the CMGND model, as well as the density curves of each mixture component. This visualization helps in understanding how each component contributes to the model and provides insights into the data distribution.

Value

A plot illustrating the marginal density along with the densities of the individual mixture components for the given data 'x'.

See Also

'cmgnd()' for estimating the model parameters.

sim_cmgnd: Function to Simulate Univariate Constrained Mixtures of Generalized Normal Distributions

Description

Simulate univariate constrained mixture of generalized normal distribution models. Remeber to set the set.seed() before the function sim_cmgnd().

Usage

sim_cmgnd(
  n = 1000,
  pi = rep(0.5, 2),
  mu = c(1, 5),
  sigma = c(1, 1),
  nu = c(2, 2)
)

Arguments

Value