Author: James Thorson

# Spatial diffusion

#### James Thorson

## Approximating diffusive movement among adjacent spatial strata

`dsem` can be specified to represent diffusion among variables that represent spatial strata where some strata are adjacent to one another.

To show this, we simulate data for five spatial strata (A, B, C, D, E) that are adjacent to one another along a single line (e.g., C is adjacent to B and D):

    library(igraph)
    library(Matrix)

    # Simulation
    adjacency_graph = make_graph( ~ A - B - C - D - E )
    A = as.matrix( adjacency_graph )

    # Diffusion rate
    Dprime = 1 * A
    diag(Dprime) = -1 * rowSums(Dprime)

    # Movement transition matrix
    M = expm( Dprime )

    # set seed for reproducibility
    set.seed(101)

    # Simulate densities
    n_times = 100
    n_burnin = 100
    x_ti = matrix( NA, nrow=n_times+n_burnin, ncol = nrow(M) )
    x_ti[1,] = rnorm(n=nrow(M), mean = 0, sd = 1 )
    for( t in 2:nrow(x_ti) ){
      x_ti[t,] = (x_ti[t-1,] %*% M)[1,] + rnorm(n=nrow(M), mean = 0, sd = 0.1)
    }

    # Subset to times after burn-in
    x_ti = x_ti[ n_burnin+seq_len(n_times), ]

We then specify a SEM that approximates diffusive movement, specifically using a diffusion-enhanced spatio-temporal process:

    library(dsem)

    # Specify SEM
    sem = "
      # Spatial correlation
      A -> B, 0, d0
      B -> C, 0, d0
      C -> D, 0, d0
      D -> E, 0, d0
      E -> D, 0, d0
      D -> C, 0, d0
      C -> B, 0, d0
      B -> A, 0, d0

      # Spatio-temporal diffusion
      A -> B, 1, d
      B -> C, 1, d
      C -> D, 1, d
      D -> E, 1, d
      E -> D, 1, d
      D -> C, 1, d
      C -> B, 1, d
      B -> A, 1, d

      # Self-limitation
      A -> A, 1, rho
      B -> B, 1, rho
      C -> C, 1, rho
      D -> D, 1, rho
      E -> E, 1, rho
    "

    # Fit
    colnames(x_ti) = c("A","B","C","D","E")
    fit = dsem(
      tsdata = ts(x_ti),
      sem = sem
    )
    #>   Coefficient_name Number_of_coefficients   Type
    #> 1           beta_z                      8  Fixed
    #> 2             mu_j                      5 Random

Finally, we can predict movement resulting from the estimated path coefficients:

    # Calculate total effect
    effect = total_effect(fit, n_lags = 3)

    # Calculate predicted movement
    Mhat = array( subset(effect,lag==1)$total_effect, dim(M) )
    dimnames(Mhat) = dimnames(M)

    # Display predicted movement
    knitr::kable( Mhat, digits=2)

| A | B | C | D | E
--- | --- | --- | --- | --- | ---
A | 0.45 | 0.29 | 0.00 | 0.00 | 0.00
B | 0.29 | 0.45 | 0.29 | 0.00 | 0.00
C | 0.00 | 0.29 | 0.45 | 0.29 | 0.00
D | 0.00 | 0.00 | 0.29 | 0.45 | 0.29
E | 0.00 | 0.00 | 0.00 | 0.29 | 0.45

And we can compare this with the true transition matrix from diffusive movement

    knitr::kable( as.matrix(M), digits=2)

| A | B | C | D | E
--- | --- | --- | --- | --- | ---
A | 0.52 | 0.31 | 0.12 | 0.04 | 0.01
B | 0.31 | 0.34 | 0.22 | 0.10 | 0.04
C | 0.12 | 0.22 | 0.31 | 0.22 | 0.12
D | 0.04 | 0.10 | 0.22 | 0.34 | 0.31
E | 0.01 | 0.04 | 0.12 | 0.31 | 0.52

Finally, we can simplify the SEM by either: * turning off the spatial correlation (i.e., simultaneous spatial interactions), such that movement is limited to one stratum per time.