Help for package leptokurticMixture

Title:

Implements Parsimonious Finite Mixtures of Multivariate Elliptical Leptokurtic-Normals

Version:

1.1

Description:

A way to fit Parsimonious Finite Mixtures of Multivariate Elliptical Leptokurtic-Normals. Two methods of estimation are implemented.

Date:

2023-09-08

Encoding:

UTF-8

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

Imports:

stats

NeedsCompilation:

yes

RoxygenNote:

7.2.3

Packaged:

2023-09-09 11:23:52 UTC; ryanbrowne

Author:

Ryan Browne [aut, cre] (0000-0003-4543-0218), Luca Bagnato [ctb], Antonio Punzo [ctb]

Maintainer:

Ryan Browne <rpbrowne@uwaterloo.ca>

Repository:

CRAN

Date/Publication:

2023-09-09 12:00:02 UTC

EM for the finite mixtures of MLN

Description

Performs a number of iterations of the EM for the multivariate elliptical leptokurtic-normal (MLN) distribution until the tolerance for the lack progress or the maximum number of iterations is reached. An implementation of parsimonious clustering models via the eigen-decomposition of the scatter matrix and allowing the concentration parameter to be varying, equal or fixed across components.

Usage

EM(
  data = NULL,
  G = 2,
  model = NULL,
  kml = c(1, 0, 1),
  n = 10,
  epsilon = 0.01,
  gpar0 = NULL,
  estimation = 1,
  label = NULL
)

Arguments

data

A n x p matrix of observations.

G

A integer determine the number of components of the mixture model.

model

a character of length 4 such as "VVVV", indicating the model; the covariance and beta parameters. The 1st position controls, lambda, the volume; "V" varying across components or "E" equal across components. The 2nd position controls the eigenvalues; V" varying across components, "E" equal across components or "I" the identity matrix. The 3rd position controls the orientation; "V" varying across components, "E" equal across components or "I" the identity matrix. The 4th position controls the concentration, beta; "V" varying across components, "E" equal across components, "F" fixed at the maximum value.

kml

a vector of length 3 indicating, the number of k-means starts, number of random starts and the number of EM iterations used for each start

n

The maximum number of EM iterations.

epsilon

The tolerance for the stopping rule; lack of progress. The default is 1e-6 but it depends on the dataset.

gpar0

A list of model parameters .

estimation

If 1 (default) use the fixed point iterations and if 2 the MM algorithm.

label

If NULL then the data has no known groups. If is.integer then some of the observations have known groups. If label[i]=k then observation belongs to group k. If label[i]=0 then observation has no known group.

Value

A list with following items

loglik - A vector of the loglikelihood values
gpar - A list containing the parameters values
z - A n x G matrix of the posterior probabilities
map - A vector the maximum a posteriori derived from z
label - The input provided.
numpar - The number of free parameters in the fitted model.
maxLoglik - The largest value from loglik.

Examples

x1 = rmln(n=100, d=4, mu=rep(5,4), diag(4), beta=2)
x2 = rmln(n=100, d=4, mu=rep(-5,4), diag(4), beta=2)
x = rbind( x1,x2)
mlnFit = EM(data=x, G=2, model="VVVF")

Compare the two methods of estimation

Description

Compare the two methods of estimation for fitting a finite mixture of multivariate elliptical leptokurtic-normal distributions; fixed point iterations and MM algorithm.

Usage

compareEstimation(
  mod = NULL,
  data = NULL,
  G = NULL,
  n = 10^4,
  tol = 1e-06,
  wt = NULL,
  n0 = 25,
  lab = NULL
)

Arguments

mod

A character of length 4 such as "VVVV", indicating the model; the covariance and beta parameters.

data

A n x p matrix of observations.

G

The number of components to fit.

n

The maximum number of EM iterations.

tol

The tolerance for the stopping rule; lack of progress. The default is 1e-6 but it depends on the dataset.

wt

a (n x d) matrix of weights for initialization if NULL, then a random weight matrix is generated.

n0

Given wt, the number of iterations used to obtain the initial parameters

lab

Using given labels (lab) as starting values.

Value

A vector of times, number of iterations and log-likelihood values.

Parsimonious model-based clustering with the multivariate elliptical leptokurtic-normal

Description

Performs parsimonious clustering with the multivariate elliptical leptokurtic-normal (MLN). There are 14 possible scale matrix structure and 2 for the kurtosis parameter for a total of 28 models.

Usage

pmln(
  data = NULL,
  G = 1:3,
  covModels = NULL,
  betaModels = "B",
  kml = c(1, 0, 1),
  label = NULL,
  scale.data = TRUE,
  veo = FALSE,
  iterMax = 1000,
  tol = 1e-08,
  pprogress = FALSE,
  method = "FP"
)

Arguments

data

A n x p matrix of observations.

G

A integer determine the number of components of the mixture model.

covModels

if NULL fit 14 possible scale matrix structures. Otherwise a character vector where each element has length 3. e.g. c("VVV", "EEE") A character of length 4 such as "VVVV", indicating the model; the covariance and beta parameters. The 1st position controls, lambda, the volume; "V" varying across components or "E" equal across components. The 2nd position controls the eigenvalues; V" varying across components, "E" equal across components or "I" the identity matrix. The 3rd position controls the orientation; "V" varying across components, "E" equal across components or "I" the identity matrix.

betaModels

set to "V", "E", "B", "F". "V" varying across components, "E" equal across components, "B" consider both "V" & "E", "F" fixed at the maximum value.

kml

a vector of length 3 indicating, the number of k-means starts, number of random starts and the number of EM iterations used for each start

label

scale.data

Should the data be scaled before clustering. The default is TRUE.

veo

"Variables exceed observations". If TRUE, fit the model even though the number variables in the model exceeds the number of observations.

iterMax

The maximum number of EM iterations for each model fitted.

tol

The tol for the stopping rule; lack of progress. The default is 1e-6 but it depends on the data set.

pprogress

If TRUE print the progress of the function.

method

If FP use the fixed point iteration method otherwise if MM use the MM method.

Value

A list of

startobject - A statement on how the models were initialized
gpar - A list of parameter values for the model choosen by the BIC
loglik - A vector of the log-likelihoods values
z - A n x G matrix of the posterior probabilities from the model choosen by the BIC
map - A vector the maximum a posteriori derived from z
BIC - An array with dimensions (G, number of fitted models, 3). The last dimension indices the loglik, number of free parameters and BIC for each fitted model.
bicModel - Information as list on the model choosen by the BIC.

Examples

x1 = rmln(n=100, d=4, mu=rep(5,4), diag(4), beta=2)
x2 = rmln(n=100, d=4, mu=rep(-5,4), diag(4), beta=2)
x = rbind( x1,x2)
mlnFit = pmln(data=x, G=2, covModels=c("VVV", "EEE"), betaModels="B")

Generate realizations from the multivariate elliptical leptokurtic-normal distribution

Description

This function calculates the log cumulative density function for the multivariate-t with scale matrix equal to the identity matrix. It finds the mode and then uses Gaussian quadrature to estimate the integral.

Usage

rmln(n = NULL, d = NULL, mu = NULL, Sigma = NULL, beta = NULL)

Arguments

n

number of observations

d

the dimension of the observations

mu

location parameter of length d

Sigma

(d x d) scatter matrix

beta

the concentration parameter

Value

A (n x d) matrix of realizations

Examples

x = rmln(n=10, d=4, mu=rep(0,4), diag(4), beta=2)