# Spatial factor analysis

#### James T. Thorson

    library(tinyVAST)
    library(fmesher)
    set.seed(101)
    options("tinyVAST.verbose" = FALSE)

`tinyVAST` is an R package for fitting vector autoregressive spatio-temporal (VAST) models. We here explore the capacity to specify a spatial factor analysis, where the spatial pattern for multiple variables is described via their estimated association with a small number of spatial latent variables.

# Spatial factor analysis

We first explore the ability to specify two latent variables for five manifest variables. To start we simulate two spatial latent variables, project via a simulated loadings matrix, and then simulate a Tweedie response for each manifest variable:

    # Simulate settings
    theta_xy = 0.4
    n_x = n_y = 10
    n_c = 5
    rho = 0.8
    resid_sd = 0.5

    # Simulate GMRFs
    R_s = exp(-theta_xy * abs(outer(1:n_x, 1:n_y, FUN="-")) )
    R_ss = kronecker(X=R_s, Y=R_s)
    delta_fs = mvtnorm::rmvnorm(n_c, sigma=R_ss )

    #
    L_cf = matrix( rnorm(n_c^2), nrow=n_c )
    L_cf[,3:5] = 0
    L_cf = L_cf + resid_sd * diag(n_c)

    #
    d_cs = L_cf %*% delta_fs

Where we can inspect the simulated loadings matrix

    dimnames(L_cf) = list( paste0("Var ", 1:nrow(L_cf)),
                           paste0("Factor ", 1:ncol(L_cf)) )
    knitr::kable( L_cf,
                  digits=2, caption="True loadings")

True loadings
| Factor 1 | Factor 2 | Factor 3 | Factor 4 | Factor 5
--- | --- | --- | --- | --- | ---
Var 1 | -0.77 | 0.26 | 0.0 | 0.0 | 0.0
Var 2 | -0.66 | 1.03 | 0.0 | 0.0 | 0.0
Var 3 | -0.30 | -0.54 | 0.5 | 0.0 | 0.0
Var 4 | -0.57 | 0.34 | 0.0 | 0.5 | 0.0
Var 5 | -0.03 | -0.24 | 0.0 | 0.0 | 0.5

We then specify the model as expected by tinyVAST:

    # Shape into longform data-frame and add error
    Data = data.frame( expand.grid(species=1:n_c, x=1:n_x, y=1:n_y),
                       "var"="logn", "z"=exp(as.vector(d_cs)) )
    Data$n = tweedie::rtweedie( n=nrow(Data), mu=Data$z, phi=0.5, power=1.5 )
    mean(Data$n==0)
    #> [1] 0.03

    # make mesh
    mesh = fm_mesh_2d( Data[,c('x','y')] )

    #
    sem = "
      f1 -> 1, l1
      f1 -> 2, l2
      f1 -> 3, l3
      f1 -> 4, l4
      f1 -> 5, l5
      f2 -> 2, l6
      f2 -> 3, l7
      f2 -> 4, l8
      f2 -> 5, l9
      f1 <-> f1, NA, 1
      f2 <-> f2, NA, 1
      1 <-> 1, NA, 0
      2 <-> 2, NA, 0
      3 <-> 3, NA, 0
      4 <-> 4, NA, 0
      5 <-> 5, NA, 0
    "

    # fit model
    out = tinyVAST( space_term = sem,
               data = Data,
               formula = n ~ 0 + factor(species),
               spatial_domain = mesh,
               family = tweedie(),
               variables = c( "f1", "f2", 1:n_c ),
               space_columns = c("x","y"),
               variable_column = "species",
               time_column = "time",
               distribution_column = "dist",
               control = tinyVASTcontrol(gmrf="proj") )
    out
    #> Call:
    #> tinyVAST(formula = n ~ 0 + factor(species), data = Data, space_term = sem,
    #>     family = tweedie(), space_columns = c("x", "y"), spatial_domain = mesh,
    #>     time_column = "time", variable_column = "species", variables = c("f1",
    #>         "f2", 1:n_c), distribution_column = "dist", control = tinyVASTcontrol(gmrf = "proj"))
    #>
    #> Run time:
    #> Time difference of 1.291343 secs
    #>
    #> Family:
    #> $obs
    #>
    #> Family: tweedie
    #> Link function: log
    #>
    #>
    #>
    #>
    #> sdreport(.) result
    #>              Estimate Std. Error
    #> alpha_j    0.07570783 0.31850774
    #> alpha_j   -0.02014888 0.39763412
    #> alpha_j    0.22318277 0.21847161
    #> alpha_j    0.14728087 0.27057852
    #> alpha_j   -0.26514652 0.14638317
    #> theta_z    0.68016356 0.11510781
    #> theta_z    0.68285927 0.15773444
    #> theta_z    0.31701846 0.10358197
    #> theta_z    0.52123769 0.10914344
    #> theta_z    0.14819781 0.09200614
    #> theta_z    0.51873790 0.13709521
    #> theta_z   -0.31998633 0.10049164
    #> theta_z    0.23601680 0.10640963
    #> theta_z   -0.21613586 0.09652980
    #> log_sigma -0.52205331 0.06761637
    #> log_sigma  0.21851154 0.13313019
    #> log_kappa -0.26761196 0.21030826
    #> Maximum gradient component: 0.002233942
    #>
    #> Proportion conditional deviance explained:
    #> [1] 0.551999
    #>
    #> space_term:
    #>    heads to from parameter start   Estimate  Std_Error   z_value      p_value
    #> 1      1  1   f1         1  <NA>  0.6801636 0.11510781  5.908926 3.443444e-09
    #> 2      1  2   f1         2  <NA>  0.6828593 0.15773444  4.329170 1.496721e-05
    #> 3      1  3   f1         3  <NA>  0.3170185 0.10358197  3.060556 2.209261e-03
    #> 4      1  4   f1         4  <NA>  0.5212377 0.10914344  4.775713 1.790719e-06
    #> 5      1  5   f1         5  <NA>  0.1481978 0.09200614  1.610738 1.072368e-01
    #> 6      1  2   f2         6  <NA>  0.5187379 0.13709521  3.783778 1.544654e-04
    #> 7      1  3   f2         7  <NA> -0.3199863 0.10049164 -3.184209 1.451504e-03
    #> 8      1  4   f2         8  <NA>  0.2360168 0.10640963  2.218002 2.655468e-02
    #> 9      1  5   f2         9  <NA> -0.2161359 0.09652980 -2.239058 2.515211e-02
    #> 10     2 f1   f1         0     1  1.0000000         NA        NA           NA
    #> 11     2 f2   f2         0     1  1.0000000         NA        NA           NA
    #> 12     2  1    1         0     0  0.0000000         NA        NA           NA
    #> 13     2  2    2         0     0  0.0000000         NA        NA           NA
    #> 14     2  3    3         0     0  0.0000000         NA        NA           NA
    #> 15     2  4    4         0     0  0.0000000         NA        NA           NA
    #> 16     2  5    5         0     0  0.0000000         NA        NA           NA
    #>
    #> Fixed terms:
    #>                     Estimate Std_Error     z_value    p_value
    #> factor(species)1  0.07570783 0.3185077  0.23769543 0.81211733
    #> factor(species)2 -0.02014888 0.3976341 -0.05067191 0.95958696
    #> factor(species)3  0.22318277 0.2184716  1.02156419 0.30698721
    #> factor(species)4  0.14728087 0.2705785  0.54431841 0.58622238
    #> factor(species)5 -0.26514652 0.1463832 -1.81131833 0.07009159

We can compare the true loadings (rotated to optimize comparison):

    Lrot_cf = rotate_pca( L_cf )$L_tf
    dimnames(Lrot_cf) = list( paste0("Var ", 1:nrow(Lrot_cf)),
                           paste0("Factor ", 1:ncol(Lrot_cf)) )
    knitr::kable( Lrot_cf,
                  digits=2, caption="Rotated true loadings")

Rotated true loadings
| Factor 1 | Factor 2 | Factor 3 | Factor 4 | Factor 5
--- | --- | --- | --- | --- | ---
Var 1 | -0.71 | 0.34 | 0.00 | -0.11 | -0.19
Var 2 | -1.20 | -0.18 | 0.00 | -0.18 | 0.12
Var 3 | 0.22 | 0.73 | -0.17 | -0.09 | 0.10
Var 4 | -0.70 | 0.25 | 0.06 | 0.37 | 0.03
Var 5 | 0.17 | 0.23 | 0.47 | -0.08 | 0.03

with the estimated loadings

    # Extract and rotate estimated loadings
    Lhat_cf = matrix( 0, nrow=n_c, ncol=2 )
    Lhat_cf[lower.tri(Lhat_cf,diag=TRUE)] = as.list(out$sdrep, what="Estimate")$theta_z
    Lhat_cf = rotate_pca( L_tf=Lhat_cf, order="decreasing" )$L_tf
    #> Warning in sqrt(Eigen$values): NaNs produced

Where we can compared the estimated and true loadings matrices:

    dimnames(Lhat_cf) = list( paste0("Var ", 1:nrow(Lhat_cf)),
                           paste0("Factor ", 1:ncol(Lhat_cf)) )
    knitr::kable( Lhat_cf,
                  digits=2, caption="Rotated estimated loadings" )

Rotated estimated loadings
| Factor 1 | Factor 2
--- | --- | ---
Var 1 | 0.64 | -0.23
Var 2 | 0.82 | 0.26
Var 3 | 0.19 | -0.41
Var 4 | 0.57 | 0.05
Var 5 | 0.07 | -0.25

Or we can specify the model while ensuring that residual spatial variation is also captured:

    #
    sem = "
      f1 -> 1, l1
      f1 -> 2, l2
      f1 -> 3, l3
      f1 -> 4, l4
      f1 -> 5, l5
      f2 -> 2, l6
      f2 -> 3, l7
      f2 -> 4, l8
      f2 -> 5, l9
      f1 <-> f1, NA, 1
      f2 <-> f2, NA, 1
      1 <-> 1, sd_resid
      2 <-> 2, sd_resid
      3 <-> 3, sd_resid
      4 <-> 4, sd_resid
      5 <-> 5, sd_resid
    "

    # fit model
    out = tinyVAST( space_term = sem,
               data = Data,
               formula = n ~ 0 + factor(species),
               spatial_domain = mesh,
               family = list( "obs"=tweedie() ),
               variables = c( "f1", "f2", 1:n_c ),
               space_columns = c("x","y"),
               variable_column = "species",
               time_column = "time",
               distribution_column = "dist",
               control = tinyVASTcontrol(gmrf="proj") )

    # Extract and rotate estimated loadings
    Lhat_cf = matrix( 0, nrow=n_c, ncol=2 )
    Lhat_cf[lower.tri(Lhat_cf,diag=TRUE)] = as.list(out$sdrep, what="Estimate")$theta_z
    #> Warning in Lhat_cf[lower.tri(Lhat_cf, diag = TRUE)] = as.list(out$sdrep, :
    #> number of items to replace is not a multiple of replacement length
    Lhat_cf = rotate_pca( L_tf=Lhat_cf, order="decreasing" )$L_tf
    #> Warning in sqrt(Eigen$values): NaNs produced

Where we can again compared the estimated and true loadings matrices:

    dimnames(Lhat_cf) = list( paste0("Var ", 1:nrow(Lhat_cf)),
                           paste0("Factor ", 1:ncol(Lhat_cf)) )
    knitr::kable( Lhat_cf,
                  digits=2, caption="Rotated estimated loadings with full rank" )

Rotated estimated loadings with full rank
| Factor 1 | Factor 2
--- | --- | ---
Var 1 | 0.69 | -0.18
Var 2 | 0.74 | 0.23
Var 3 | 0.07 | -0.42
Var 4 | 0.47 | -0.02
Var 5 | 0.07 | -0.07