# GFM: alternate maximization and information criterion

#### Wei Liu

#### 2026-01-17

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)

## Session information

    sessionInfo()
    #> R version 4.4.1 (2024-06-14 ucrt)
    #> Platform: x86_64-w64-mingw32/x64
    #> Running under: Windows 11 x64 (build 26100)
    #>
    #> Matrix products: default
    #>
    #>
    #> locale:
    #> [1] LC_COLLATE=C
    #> [2] LC_CTYPE=Chinese (Simplified)_China.utf8
    #> [3] LC_MONETARY=Chinese (Simplified)_China.utf8
    #> [4] LC_NUMERIC=C
    #> [5] LC_TIME=Chinese (Simplified)_China.utf8
    #>
    #> time zone: Asia/Shanghai
    #> tzcode source: internal
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] digest_0.6.37     R6_2.5.1          fastmap_1.2.0     xfun_0.47
    #>  [5] cachem_1.1.0      knitr_1.48        htmltools_0.5.8.1 rmarkdown_2.28
    #>  [9] lifecycle_1.0.4   cli_3.6.3         sass_0.4.9        jquerylib_0.1.4
    #> [13] compiler_4.4.1    rstudioapi_0.16.0 tools_4.4.1       evaluate_1.0.0
    #> [17] bslib_0.8.0       yaml_2.3.10       rlang_1.1.4       jsonlite_1.8.9