cwm {flexCWM}

Fit for the CWM

Description

Maximum likelihood fitting of the cluster-weighted model by the EM algorithm.

Usage

cwm(formulaY = NULL, familyY = gaussian, data, Xnorm = NULL, Xbin = NULL,
  Xpois = NULL, Xmult = NULL, modelXnorm = NULL, Xbtrials = NULL, k = 1:3, 
  initialization = c("random.soft", "random.hard", "kmeans", "mclust", "manual"), 
  start.z = NULL, seed = NULL, maxR = 1, iter.max = 1000, threshold = 1.0e-04, 
  eps = 1e-100, parallel = FALSE,   pwarning = FALSE)

Arguments

formulaY

an optional object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.

familyY

a description of the error distribution and link function to be used for the conditional distribution of Y in each mixture component. This can be a character string naming a family function, a family function or the result of a call to a family function. The following family functions are supported:

• binomial(link = "logit")

• gaussian(link = "identity")

• Gamma(link = "log")

• inverse.gaussian(link = "1/mu^2")

• poisson(link = "log")

• student.t(link = "identity")

Default value is gaussian(link = "identity").

data

an optional data.frame, list, or environment with the variables needed to use formulaY.

Xnorm, Xbin, Xpois, Xmult

an optional matrix containing variables to be used for marginalization having normal, binomial, Poisson and multinomial distributions.

modelXnorm

an optional vector of character strings indicating the parsimonious models to be fitted for variables in Xnorm. The default is c("E", "V") for a single continuous covariate, and c("EII", "VII", "EEI", "VEI", "EVI", "VVI", "EEE", "VEE", "EVE", "EEV", "VVE", "VEV", "EVV", "VVV") for multivariate continuous covariates (see mixture:gpcm for details).

Xbtrials

an optional vector containing the number of trials for each column in Xbin. If omitted, the maximum of each column in Xbin is used.

k

an optional vector containing the numbers of mixture components to be tried. Default value is 1:3.

initialization

an optional character string. It sets the initialization strategy for the EM-algorithm. It can be:

• "random.soft"

• "random.hard"

• "kmeans"

• "mclust"

• "manual"

Default value is "random.soft".

start.z

matrix of soft or hard classification: it is used only if initialization = "manual".Only models with the same number of mixture components as the columns of this matrix will be fit.

seed

an optional scalar. It sets the seed for the random number generator, when random initializations are used; if NULL, current seed is not changed. Default value is NULL.

maxR

number of initializations to be tried. Default value is 1.

iter.max

an optional scalar. It sets the maximum number of iterations in the EM-algorithm. Default value is 200.

threshold

an optional scalar. It sets the threshold for the Aitken acceleration procedure. Default value is 1.0e-04.

eps

an optional scalar. It sets the smallest value for eigenvalues of covariance matrices for Xnorm. Default value is 1e-100.

parallel

When TRUE, the package parallel is used for parallel computation. When several models are estimated, computational time is reduced. The number of cores to use may be set with the global option cl.cores; default value is detected using detectCores().

pwarning

When TRUE, warnings are printed.

Details

When familyY = binomial, the response variable must be a matrix with two columns, where the first column is the number of "successes" and the second column is the number of "failures". When several models have been estimated, methods summary and print consider the best model according to the information criterion in criterion, among the estimated models having a number of components among those in k an error distribution among those in familyY and a parsimonious model among those in modelXnorm.

Value

This function returns a class cwm object, which is a list of values related to the model selected. It contains:

call

an object of class call.

formulaY

an object of class formula containing a symbolic description of the model fitted.

familyY

the distribution used for the conditional distribution of Y in each mixture component.

data

a data.frame with the variables needed to use formulaY.

concomitant

a list containing Xnorm, Xbin, Xpois, Xmult.

Xbtrials

number of trials used for Xbin.

models

a list; each element is related to one of the models fitted. Each element is a list and contains:

• posterior posterior probabilities

• iter number of iterations performed in EM algorithm

• k number of (fitted) mixture components.

• size estimated size of the groups.

• cluster classification vector

• loglik final log-likelihood value

• df overall number of estimated parameters

• prior weights for the mixture components

• IC list containing values of the information criteria

• converged logical; TRUE if EM algorithm converged

• GLModels a list; each element is related to a mixture component and contains:

  • model a "glm" class object.

  • sigma estimated local scale parameters of the conditional distribution of Y, when familyY is gaussian or student.t

  • t_df estimated degrees of freedom of the t distribution, when familyY is student.t

  • nuY estimated shape parameter, when familyY is Gamma. The gamma distribution is parameterized according to McCullagh & Nelder (1989, p. 30)

• concomitant a list with estimated concomitant variables parameters for each mixture component

  • normal.d, multinomial.d, poisson.d, binomial.d marginal distribution of concomitant variables

  • normal.mu mixture component means for Xnorm

  • normal.Sigma mixture component covariance matrices for Xnorm

  • normal.model models fitted for Xnorm

  • multinomial.probs multinomial distribution probabilities for Xmult

  • poisson.lambda lambda parameters for Xpois

  • binomial.p binomial probabilities for Xbin

Author(s)

Mazza A., Punzo A., Ingrassia S.

References

Mazza, A., Ingrassia, S., and Punzo, A. (2018). flexCWM: A Flexible Framework for Cluster-Weighted Models. Journal of Statistical Software, 86(2), 1-30.

Ingrassia, S., Minotti, S. C., and Vittadini, G. (2012). Local Statistical Modeling via the Cluster-Weighted Approach with Elliptical Distributions. Journal of Classification, 29(3), 363-401.

Ingrassia, S., Minotti, S. C., and Punzo, A. (2014). Model-based clustering via linear cluster-weighted models. Computational Statistics and Data Analysis, 71, 159-182.

Ingrassia, S., Punzo, A., and Vittadini, G. (2015). The Generalized Linear Mixed Cluster-Weighted Model. Journal of Classification, 32(forthcoming)

McCullagh, P. and Nelder, J. (1989). Generalized Linear Models. Chapman & Hall, Boca Raton, 2nd edition

Punzo, A. (2014). Flexible Mixture Modeling with the Polynomial Gaussian Cluster-Weighted Model. Statistical Modelling, 14(3), 257-291.

See Also

flexCWM-package

Examples

## an exemple with artificial data
data("ExCWM")
attach(ExCWM)
str(ExCWM)

# mixtures of binomial distributions
resXbin <- cwm(Xbin = Xbin, k = 1:2, initialization = "kmeans")
getParXbin(resXbin)

# Mixtures of Poisson distributions
resXpois <- cwm(Xpois = Xpois, k = 1:2, initialization = "kmeans")
getParXpois(resXpois)

# parsimonious mixtures of multivariate normal distributions
resXnorm <- cwm(Xnorm = cbind(Xnorm1,Xnorm2), k = 1:2, initialization = "kmeans")
getParXnorm(resXnorm)

## an exemple with real data
data("students")
attach(students)
str(students)
# CWM
fit2 <- cwm(WEIGHT ~ HEIGHT + HEIGHT.F , Xnorm = cbind(HEIGHT, HEIGHT.F), 
  k = 2, initialization = "kmeans", modelXnorm = "EEE")
summary(fit2, concomitant = TRUE)
plot(fit2)