Additional features

Model

We assume that the response variable \(Y_{it}\) for region \(i\) at time \(t\) follows a normal distribution given the partition \(M\): \[ Y_{it} \mid \mu_{it}, M \stackrel{ind}{\sim} \text{Normal}(\mu_{it}, \sigma^2), \] where the mean \(\mu_{it}\) is modeled based on the cluster assignment \(c_i\): \[ \mu_{it} = \alpha_{c_i} + f_{c_i,t}, \] where \(\alpha_{c_i}\) is the intercept for cluster \(c_i\), and \(f_{c_i,t}\) is a latent random effect modeled as a random walk process. Specifically, we impose the condition: \[ f_{c_i,t} - f_{c_i,t-1} \sim N(0, \nu^{-1}), \quad \text{for } t = 2, \dots, n. \]

The prior for the hyperparameter \(\nu_c\) is defined as: \[ \log(\nu_c) \sim \text{Normal}(-2, 1). \]

Initial clustering

sfclust uses an undirected graph to represent connections between regions and proposes spatial clusters using this graph through minimum spanning trees (MST). By default, sfclust accepts the argument graphdata, which should include:

• An undirected igraph object representing spatial connections,
• An MST of that graph, and
• A membership vector indicating the initial cluster assignments for each region.

For simplicity, you can use the genclust function to generate an initial random partitioning with a specified number of clusters. In this example, we create a partition with 20 clusters:

set.seed(123)
initial_cluster <- genclust(st_geometry(stgaus), nclust = 20)
names(initial_cluster)

#> [1] "graph"      "mst"        "membership"

Now, let’s visualize how the regions were randomly clustered:

st_sf(st_geometry(stgaus), cluster = factor(initial_cluster$membership)) |>
  ggplot() +
    geom_sf(aes(fill = cluster)) +
    theme_bw()

Sampling with sfclust

We will initiate the Bayesian clustering algorithm using our generated partition (initial_cluster). Additionally, we use the capabilites of INLA::inla to define a model that includes a random walk process in the linear predictor, and custom priors for the scale hyperparameter. To begin, we will run the algorithm for only 50 iterations.

result <- sfclust(stgaus, graphdata = initial_cluster, niter = 50,
    formula = y ~ f(idt, model = "rw1", hyper = list(prec = list(prior = "normal", param = c(-2, 1)))))
result

#> Within-cluster formula:
#> y ~ f(idt, model = "rw1", hyper = list(prec = list(prior = "normal", 
#>     param = c(-2, 1))))
#> 
#> Clustering hyperparameters:
#>      q  birth  death change  hyper 
#>  0.500  0.425  0.425  0.100  0.050 
#> 
#> Clustering movement counts:
#>  births  deaths changes  hypers 
#>      11       6       3       3 
#> 
#> Log marginal likelihood (sample 50 out of 50): 2268.291

The output indicates that after starting with 20 clusters, the algorithm created 11 new clusters (births), removed 6 clusters (deaths), changed the membership of 3 clusters, and modified the minimum spanning tree (MST) 3 times.

The plot method allows us to select which graph to produce. For example, we can visualize only the log marginal likelihood to diagnose convergence. With just 50 iterations, we can see that the log marginal likelihood has not yet achieved convergence.

plot(result, which = 2)

Continue sampling

To achieve convergence, we can continue the sampling using the update function and specify the number of new iterations with the niter argument:

result2 <- update(result, niter = 2000)
result2

#> Within-cluster formula:
#> y ~ f(idt, model = "rw1", hyper = list(prec = list(prior = "normal", 
#>     param = c(-2, 1))))
#> 
#> Clustering hyperparameters:
#>      q  birth  death change  hyper 
#>  0.500  0.425  0.425  0.100  0.050 
#> 
#> Clustering movement counts:
#>  births  deaths changes  hypers 
#>      35      50       9      88 
#> 
#> Log marginal likelihood (sample 2000 out of 2000): 12708.35

With 2000 additional iterations, there have been many clustering movements. Furthermore, when visualizing the results, we can observe that the log marginal likelihood has achieved convergence.

plot(result2, which = 2)