# 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 (Thorson et al. 2015).

# 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(), spatial_domain = mesh, control = tinyVASTcontrol(gmrf = "proj"),
    #>     space_columns = c("x", "y"), time_column = "time", variable_column = "species",
    #>     variables = c("f1", "f2", 1:n_c), distribution_column = "dist")
    #>
    #> Run time:
    #> Time difference of 2.849502 secs
    #>
    #> Family:
    #> $obs
    #>
    #> Family: tweedie
    #> Link function: log
    #>
    #>
    #>
    #>
    #> sdreport(.) result
    #>              Estimate Std. Error
    #> alpha_j    0.07569060 0.31851467
    #> alpha_j   -0.02016919 0.39764587
    #> alpha_j    0.22317503 0.21847588
    #> alpha_j    0.14727256 0.27058070
    #> alpha_j   -0.26515247 0.14638935
    #> theta_z    0.68017152 0.11511151
    #> theta_z    0.68287021 0.15773931
    #> theta_z    0.31702161 0.10358374
    #> theta_z    0.52124050 0.10914511
    #> theta_z    0.14820460 0.09200849
    #> theta_z    0.51875214 0.13709851
    #> theta_z   -0.31999054 0.10049337
    #> theta_z    0.23600834 0.10641093
    #> theta_z   -0.21614664 0.09653182
    #> log_sigma -0.52203929 0.06761656
    #> log_sigma  0.21850709 0.13313013
    #> log_kappa -0.26762174 0.21031509
    #> Maximum gradient component: 0.00254427
    #>
    #> Proportion conditional deviance explained:
    #> [1] 0.5312474
    #>
    #> space_term:
    #>    heads to from parameter start   Estimate  Std_Error   z_value      p_value
    #> 1      1  1   f1         1  <NA>  0.6801715 0.11511151  5.908805 3.445975e-09
    #> 2      1  2   f1         2  <NA>  0.6828702 0.15773931  4.329106 1.497157e-05
    #> 3      1  3   f1         3  <NA>  0.3170216 0.10358374  3.060535 2.209422e-03
    #> 4      1  4   f1         4  <NA>  0.5212405 0.10914511  4.775665 1.791141e-06
    #> 5      1  5   f1         5  <NA>  0.1482046 0.09200849  1.610771 1.072297e-01
    #> 6      1  2   f2         6  <NA>  0.5187521 0.13709851  3.783791 1.544575e-04
    #> 7      1  3   f2         7  <NA> -0.3199905 0.10049337 -3.184196 1.451569e-03
    #> 8      1  4   f2         8  <NA>  0.2360083 0.10641093  2.217896 2.656195e-02
    #> 9      1  5   f2         9  <NA> -0.2161466 0.09653182 -2.239123 2.514790e-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  0.07569060 0.3185147  0.23763614 0.81216332
    #> factor(species)2 -0.02016919 0.3976459 -0.05072149 0.95954745
    #> factor(species)3  0.22317503 0.2184759  1.02150879 0.30701345
    #> factor(species)4  0.14727256 0.2705807  0.54428332 0.58624652
    #> factor(species)5 -0.26515247 0.1463893 -1.81128250 0.07009713
    #>
    #> Sanity check:
    #>
    #> **Possible issues detected! Check output of sanity().**

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

Runtime for this vignette: 10.27 secs

# Works cited

Thorson, Mark D. Scheuerell, Andrew O. Shelton, Kevin E. See, Hans J. Skaug, and Kasper Kristensen. 2015. “Spatial Factor Analysis: A New Tool for Estimating Joint Species Distributions and Correlations in Species Range.” Methods in Ecology and Evolution 6 (6): 627–37. https://doi.org/10.1111/2041-210X.12359.