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())

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

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’.

Author

Kei Hirose
mail@keihirose.com

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

References

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

Examples

Run this code

#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)

# \donttest{
#plot solution path
plot(fit)
# }

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