GFM: alternate maximization and information criterion'
In GFM: Generalized Factor Model

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

In this tutorial, we show that the alternate maximization (AM) is used in the first step of the two-step estimation method and the information criterion (IC) method is adopted to choose the number of factors.

Fit GFM model using simulated data

The package can be loaded with the command:

library("GFM")
set.seed(1) # set a random seed for reproducibility.

GFM can handle data with homogeneous normal variables

First, we generate the data with homogeneous normal variables.

## Homogeneous  normal variables
  dat <- gendata(q = 2, n=100, p=100, rho=3)

Then, we set the algorithm parameters and fit model

# Obtain the observed data
  XList <- dat$XList # this is the data in the form of matrix list.
  str(XList)
  X <- dat$X # this is the data in form of matrix
# set variables' type, 'gaussian' means there is  continous variable type.
  types <- 'gaussian'

Third, we fit the GFM model with user-specified number of factors.

# specify q=2
  gfm1 <- gfm(XList, types, algorithm="AM", q=2, verbose = FALSE)

  # measure the performance of GFM estimators in terms of canonical correlations
  measurefun(gfm1$hH, dat$H0, type='ccor')
  measurefun(gfm1$hB, dat$B0, type='ccor')

The number of factors can also be determined by data-driven manners.

# select q automatically
  hq <- chooseFacNumber(XList, types, select_method='IC', q_set = 1:6, verbose = FALSE, parallelList=list(parallel=TRUE))
  hq

GFM outperforms LFM in analyzing data with heterogeous normal variables

First, we generate the data with heterogeous normal variables and set the parameters of algorithm.

  dat <- gendata(seed=1, n=100, p=100, type='heternorm', q=2, rho=1)
 # Obtain the observed data
  XList <- dat$XList # this is the data in the form of matrix list.
  str(XList)
  X <- dat$X # this is the data in form of matrix
# set variables' type, 'gaussian' means there is  continous variable type.
  types <- 'gaussian'

Third, we fit the GFM model with user-specified number of factors and compare the results with that of linear factor models.

# specify q=2
  gfm1 <- gfm(XList, types,  algorithm="AM", q=2, verbose = FALSE)

  # measure the performance of GFM estimators in terms of canonical correlations
  corH_gfm <- measurefun(gfm1$hH, dat$H0, type='ccor')
  corB_gfm <- measurefun(gfm1$hB, dat$B0, type='ccor')

  lfm1 <- Factorm(X, q=2)
  corH_lfm <- measurefun(lfm1$hH, dat$H0, type='ccor')
  corB_lfm <- measurefun(lfm1$hB, dat$B0, type='ccor')

  library(ggplot2)
  df1 <- data.frame(CCor= c(corH_gfm, corH_lfm, corB_gfm, corB_lfm),
                    Method =factor(rep(c('GFM', "LFM"), times=2)),
                    Quantity= factor(c(rep('factors',2), rep("loadings", 2))))
  ggplot(data=df1, aes(x=Quantity, y=CCor, fill=Method)) + geom_bar(position = "dodge", stat="identity",width = 0.5)

The number of factors can also be determined by data-driven manners.

# select q automatically
  hq <- chooseFacNumber(XList, types, select_method='IC', q_set = 1:6, verbose = FALSE, parallelList=list(parallel=TRUE))

GFM outperforms LFM in analyzing data with Count(Poisson) variables

First, we generate the data with Count(Poisson) variables and set the parameters of algorithm.

  q <- 3; p <- 200
  dat <- gendata(seed=1, n=200, p=p, type='pois', q=q, rho=4)
  # Obtain the observed data
  XList <- dat$XList # this is the data in the form of matrix list.
  str(XList)
  X <- dat$X # this is the data in form of matrix
# set variables' type, 'gaussian' means there is  continous variable type.
  types <- 'poisson'

Second, we we fit the GFM models given the true number of factors.

  system.time(
   gfm1 <- gfm(XList, types,  algorithm="AM", q=3, verbose = FALSE)
  )

system.time(
  hq <- chooseFacNumber(XList, types, q_set=1:6, select_method = "IC", parallelList=list(parallel=TRUE))
)

Third, we compare the results with that of linear factor models.

  # measure the performance of GFM estimators in terms of canonical correlations
  corH_gfm <- measurefun(gfm1$hH, dat$H0, type='ccor')
  corB_gfm <- measurefun(gfm1$hB, dat$B0, type='ccor')

  lfm1 <- Factorm(X, q=3)
  corH_lfm <- measurefun(lfm1$hH, dat$H0, type='ccor')
  corB_lfm <- measurefun(lfm1$hB, dat$B0, type='ccor')

  library(ggplot2)
  df1 <- data.frame(CCor= c(corH_gfm, corH_lfm, corB_gfm, corB_lfm),
                    Method =factor(rep(c('GFM', "LFM"), times=2)),
                    Quantity= factor(c(rep('factors',2), rep("loadings", 2))))
  ggplot(data=df1, aes(x=Quantity, y=CCor, fill=Method)) + geom_bar(position = "dodge", stat="identity",width = 0.5)

GFM outperforms LFM in analyzing data with the mixed-types of count and categorical variables

First, we generate the data with Count(Poisson) variables and set the parameters of algorithm. Then fit the GFM model with user-specified number of factors.

  dat <- gendata(seed=1, n=200, p=200, type='pois_bino', q=2, rho=2)
  # Obtain the observed data
  XList <- dat$XList # this is the data in the form of matrix list.
  str(XList)
  X <- dat$X # this is the data in form of matrix
  # set variables' type, 'gaussian' means there is  continous variable type.
  types <- dat$types
  table(dat$X[,1])
  table(dat$X[, 200])
  # user-specified q=2
  gfm2 <- gfm(XList, types,  algorithm="AM", q=2, verbose = FALSE)
  measurefun(gfm2$hH, dat$H0, type='ccor')
  measurefun(gfm2$hB, dat$B0, type='ccor')

Third, we compare the results with that of linear factor models.

  #  select q automatically
  hq <- chooseFacNumber(XList, types, select_method='IC', q_set = 1:4, verbose = FALSE, parallelList=list(parallel=TRUE))
  # measure the performance of GFM estimators in terms of canonical correlations
  corH_gfm <- measurefun(gfm2$hH, dat$H0, type='ccor')
  corB_gfm <- measurefun(gfm2$hB, dat$B0, type='ccor')

Compare with linear factor models

  lfm1 <- Factorm(dat$X, q=3)
  corH_lfm <- measurefun(lfm1$hH, dat$H0, type='ccor')
  corB_lfm <- measurefun(lfm1$hB, dat$B0, type='ccor')

  library(ggplot2)
  df1 <- data.frame(CCor= c(corH_gfm, corH_lfm, corB_gfm, corB_lfm),
                    Method =factor(rep(c('GFM', "LFM"), times=2)),
                    Quantity= factor(c(rep('factors',2), rep("loadings", 2))))
  ggplot(data=df1, aes(x=Quantity, y=CCor, fill=Method)) + geom_bar(position = "dodge", stat="identity",width = 0.5)