fanc {fanc}

fanc (penalized maximum likelihood factor analysis via nonconvex penalties)

Description

This package computes the solution path of penalized maximum likelihood estimates via MC+ penalties.

Usage

fanc(x, factors, n.obs, rho, gamma, cor.factor=FALSE, normalize=TRUE, 
     normalize.penalty=FALSE, covmat, type="MC", model="FA", control=list())

Arguments

x

A data matrix.

factors

The number of factors.

cor.factor

An indicator of the factor correlation. If "TRUE", the factor correlation is considered. Default is "FALSE".

normalize

If "TRUE", each variable is normalized, otherwise it is left alone.

normalize.penalty

If "TRUE", the penalty term for each variable has a weight so that the loading matrix is normalized.

rho

The values of rho. It can be a scalar or a matrix.

gamma

The values of gamma. It must be a vector.

covmat

A covariance matrix, which is needed if the data matrix "x" is not available.

n.obs

The number of observations, which is needed to calculate the model selection criteria and goodness-of-fit indices when the data matrix "x" is not available.

type

Type of penalty. If "MC", the MC penalty is used. If "prenet", the prenet penalty is used. If "enet", the elastic penalty is used. Default is "MC".

model

Type of model. "FA", the factor analysis model is used. If "PPCA", the probabilistic principal component analysis is conducted. In the PPCA, the unique variances have the same value. Default is "FA".

control

A list of control parameters. See ‘Details’.

Details

The control argument is a list that can supply any of the following components:

length.rho

Candidates of tuning parameters which is used for grid search of reparametrization of MC+.

length.gamma

A length of tuning parameter which controls sparsenesses. For each rho, gamma=Inf yields soft threshold operator (i.e., lasso penalty) and gamma=+1 produces hard threshold operator.

max.rho

Maximum value of rho.

max.gamma

A maximum value of gamma (excludes Inf.).

min.gamma

A minimum value of gamma.

eta

A tuning parameter used for preventing the occurrence of improper solutions. eta must be non-negative.

ncand.initial

The number of candidates of initial values of factor loadings.

ncand.initial.prenet

The number of candidates of initial values for prenet penalty. Because the prenet penalty is unstable when rho is large, ncand.initial.prenet must be large. Default is 1000.

maxit.em

A maximum number of iterations for EM algortihm.

maxit.cd

A maximum number of iterations for coordinate descent algortihm.

maxit.bfgs

A maximum number of iterations for BFGS algorithm used in the update of factor correlation.

maxit.initial

A maximum number of iterations for choosing the initial values.

start

Type of start. If "cold", the initial value of factor loadings is randomly chosen for each tuning parameter, which can be slow.

Delta

A proportion of maximum value of rho to minimum value of rho, i.e., rho.min =Delta*rho.max.

min.uniquevar

A minimum value of unique variances.

tol.em

A positive scalar giving the tolerance at which the parameter in EM is considered close enough to zero to terminate the algorithm.

tol.cd

A positive scalar giving the tolerance at which the factor loadings in coordinate descent is considered close enough to zero to terminate the algorithm.

tol.bfgs

A positive scalar giving the tolerance at which the factor correlation in BFGS algorithm is considered close enough to zero to terminate the algorithm.

min.rhozero

If "TRUE", the minimum value of "rho" is zero.

zita

A value of hyper-parameter of factor correlation.

progress

If "TRUE", the progress for each tuning parameter is displayed.

openmp

If "TRUE", the parallel computation via OpenMP is excecuted.

num.threads

The number of threads of the openmp. Only used when openmp is "TRUE",

gamma.ebic

The value of gamma used in the extended BIC

Value

loadings

factor loadings

uniquenesses

unique variances

Phi

factor correlation

rho

rho

AIC

AIC

BIC

BIC

CAIC

CAIC

df

degrees of freedom (number of non-zero parameters for the lasso estimation)

criteria

values of AIC, BIC and CAIC

goodness.of.fit

values of GFI and AGFI

gamma

a value of gamma

Npflag

If the number of observation is larger than the number of variables, 1, otherwise 0.

factors

the number of factors

cor.factor

An indicator of the factor correlation

x

data matrix

convergence

indicator of convergence of EM algorithm, coordinate descent and BFGS. If all of these variables are 0, the algorithm has been converged

Author(s)

Kei Hirose
[e-mail]

References

Hirose, K. and Yamamoto, M. (2014). Sparse estimation via nonconcave penalized likelihood in a factor analysis model,
Statistics and Computing, in press

See Also

out and plot.fanc objects.

Examples

#generate data
set.seed(0)
loadings0 <- matrix(c(rep(0.8,5),rep(0,5),rep(0,5),rep(0.8,5)),10,2)
common.factors0 <- matrix(rnorm(50*2),50,2)
unique.factors0 <- matrix(rnorm(50*10,sd=sqrt(0.36)),50,10)
x <- common.factors0 %*% t(loadings0) + unique.factors0

#fit data
fit <- fanc(x,2)
fit2 <- fanc(x,2,cor.factor=TRUE) #factor correlation is estimated

#print candidates of gamma and rho
print(fit)

#output for fixed tuning parameters
out(fit, rho=0.1, gamma=Inf)

#select a model via model selection criterion
select(fit, criterion="BIC", gamma=Inf)


#plot solution path
plot(fit)