FactMixtAnalysis (version 1.0)

fma: Fitting Factor Mixture Analysis model by the EM algorithm

Description

fma is used to perform Factor Mixture Analysis (with covariates) on a matrix of data by the Expectation Maximization algorithm.

Usage

fma(y, k, r, x.z = NULL, x.w = NULL, it = 15, eps = 1e-04, seed = 4, scaling = FALSE, init = NULL)

Arguments

y

A data matrix with n rows representing observations and p columns representing variables.

k

The number of the mixture components.

r

The number of factors.

x.z

A matrix of covariates with n rows representing observations and q columns representing the predictors. These covariates are assumed to linearly affect the factor means.

x.w

A matrix of covariates with n rows representing observations and m columns representing the predictors. These covariates are assumed to differently affect the a priori probability of group membership.

it

The maximum number of iterations of the EM algorithm. By default it is set to 15.

eps

The lower bound for relative variation of the likelihood. It is used as alternative stopping rule for the EM algorithm: if the relative increment of the likelihood is lower than eps the fitting is stopped. The default is 0.0001.

seed

Fix the seed of the running. Default is 4.

scaling

If TRUE (FALSE is default) the data are scaled before fitting the FMA model.

init

A list containing initial values for all (of some) model parameters. If NULL (default) the algorithm starts from random values.

Value

H

The estimated factor loading matrix.

lik

The log-likelihood computed at each iteration of the EM algorithm.

w

A matrix with the estimated weights of the mixture.

Beta

An array of dimension k x r x (q +1) containing the vectors of regression coefficients which are allowed to vary across the components. When no covariate is incorporated into the model, i.e. x.z is NULL, Beta has dimension k x r and it correponds to the estimated component means of the mixture.

phi

A matrix of dimension k x m which contains the coefficients of the covariates affecting the group membership.

sigma

An array of dimension k x r x r which contains the estimated component covariance of the mixture.

psi

The noise diagonal variance matrix.

ph.y

The posterior distribution of each mixture components.

z

The reconstructed factor scores by Thomposon method.

index

The allocation vector.

bic

The BIC value.

aic

The AIC value.

elapsed

Computational time in seconds.

Details

Factor Mixture Analysis is a particular factor model with non Gaussian factors modelled by a multivariate Gaussian mixture. The p observed variables y are modelled in terms of the smaller set of r factors, z, and an additive specific term u: y=Hz+u, where u is assumed to be normally distributed with diagonal variance matrix Psi. H is the factor loading matrix. The model is fitted by the EM algorithm. The code implements also factor mixture model with covariates. Covariates may affect the observed variables into two manners: they are assumed to linearly affect the factor means (x.z) and \ or they can differently affect the a priori probability of group membership (x.w). The default is NULL which means that covariates are not incorporated in the model.

References

A. Montanari and C. Viroli (2010), Heteroscedastic Factor Mixture Analysis, Statistical Modelling, 10(4), 441-460.

A. Montanari and C. Viroli (2011), Dimensionally reduced mixtures of regression models, Journal of Statistical Planning and Inference, 141, 1744-1752.

C. Viroli (2011), Using factor mixture analysis to model heterogeneity, cognitive structure and determinants of dementia: an application to the Aging, Demographics, and Memory Study, Statistics in Medicine, to appear.

Examples

data(crabs)
y=as.matrix(crabs[,4:8])
fit=fma(y,k=2,r=1,it=50,eps=0.0001,scaling=TRUE) # without covariates
misc(fit$index,crabs[,1]) # compute the misclassification error

sex=as.double(crabs[,2])-1
fit=fma(y,k=2,r=1,x.z=sex,x.w=sex,it=50,eps=0.0001,scaling=TRUE) # with covariates