Parameter Selection

Setup

The smoothLRC method uses a smoothing parameter (\(\lambda\)) and a rank parameter (\(k\)) to produce results. This article demonstrates the selection of these parameters.

library(smoothLRC)
library(ggplot2)
library(dplyr)
set.seed(1)

Load Data

Using the spatialLIBD package, we’ll load sample 151671 from the DLPFC data.

sample <- "151671"

sce <- spatialLIBD::fetch_data(type = "sce")
sce <- sce[, sce$sample_name == sample]

Parameter Grid

Here we’ll consider \(\lambda=(5, 10, 15)\) and \(k=(10, 20)\). Further, we’ll use 100 pixels as a test size.

lambdas <- c(5, 10, 15)
ks <- c(10, 20)
parameter_grid <- expand.grid(lambda = lambdas, k = ks)
test_size <- 100

Cross-Validation

Next, we’ll use cross-validation to select the optimal parameter values.

metrics <- NULL
for(combo in 1:nrow(parameter_grid)){
  print(parameter_grid[combo,])
  combination <- smooth_cv(sce, lambda = parameter_grid[combo,"lambda"], k = parameter_grid[combo,"k"], test_size, maxiter = 5)
  metrics <- rbind(metrics, tibble(lambda = parameter_grid[combo,"lambda"], 
                                   k = parameter_grid[combo,"k"],
                                   penal_lik = combination$penal_like,
                                   cv_like = combination$cv_like)
                   )
}
#>   lambda  k
#> 1      5 10
#> [1] "Initializing components..."
#> [1] "Running smoothLRC on training set..."
#> iteration: 1 | convergence: 0.422554 | 0.0563444 | 1
#> iteration: 2 | convergence: 0.173078 | 0.0404936 | 0.00987567
#> iteration: 3 | convergence: 0.163679 | 0.0450668 | 0.0102672
#> iteration: 4 | convergence: 0.175742 | 0.0471282 | 0.0110242
#> iteration: 5 | convergence: 0.182077 | 0.0482414 | 0.0120774
#> [1] "Compute neighborhood likelihood..."
#> [1] "Done!"
#>   lambda  k
#> 2     10 10
#> [1] "Initializing components..."
#> [1] "Running smoothLRC on training set..."
#> iteration: 1 | convergence: 0.422057 | 0.0554037 | 1
#> iteration: 2 | convergence: 0.164508 | 0.0401319 | 0.00985858
#> iteration: 3 | convergence: 0.161535 | 0.0453195 | 0.010362
#> iteration: 4 | convergence: 0.186979 | 0.0486879 | 0.0112138
#> iteration: 5 | convergence: 0.184922 | 0.0508453 | 0.012346
#> [1] "Compute neighborhood likelihood..."
#> [1] "Done!"
#>   lambda  k
#> 3     15 10
#> [1] "Initializing components..."
#> [1] "Running smoothLRC on training set..."
#> iteration: 1 | convergence: 0.42013 | 0.0539799 | 1
#> iteration: 2 | convergence: 0.167513 | 0.0398524 | 0.00986839
#> iteration: 3 | convergence: 0.176619 | 0.0454105 | 0.0104753
#> iteration: 4 | convergence: 0.191349 | 0.0489113 | 0.0113943
#> iteration: 5 | convergence: 0.181778 | 0.0511128 | 0.012534
#> [1] "Compute neighborhood likelihood..."
#> [1] "Done!"
#>   lambda  k
#> 4      5 20
#> [1] "Initializing components..."
#> [1] "Running smoothLRC on training set..."
#> iteration: 1 | convergence: 0.496827 | 0.0727383 | 1
#> iteration: 2 | convergence: 0.234482 | 0.0531852 | 0.0114419
#> iteration: 3 | convergence: 0.209897 | 0.0595182 | 0.0125506
#> iteration: 4 | convergence: 0.185418 | 0.0596158 | 0.0143264
#> iteration: 5 | convergence: 0.183947 | 0.0612591 | 0.0167131
#> [1] "Compute neighborhood likelihood..."
#> [1] "Done!"
#>   lambda  k
#> 5     10 20
#> [1] "Initializing components..."
#> [1] "Running smoothLRC on training set..."
#> iteration: 1 | convergence: 0.492975 | 0.0694401 | 1
#> iteration: 2 | convergence: 0.227639 | 0.0526918 | 0.0111891
#> iteration: 3 | convergence: 0.2107 | 0.0592167 | 0.012326
#> iteration: 4 | convergence: 0.191362 | 0.0600496 | 0.0141495
#> iteration: 5 | convergence: 0.191143 | 0.0630135 | 0.0165308
#> [1] "Compute neighborhood likelihood..."
#> [1] "Done!"
#>   lambda  k
#> 6     15 20
#> [1] "Initializing components..."
#> [1] "Running smoothLRC on training set..."
#> iteration: 1 | convergence: 0.491289 | 0.0671663 | 1
#> iteration: 2 | convergence: 0.238924 | 0.052163 | 0.0110855
#> iteration: 3 | convergence: 0.201901 | 0.0582762 | 0.0122061
#> iteration: 4 | convergence: 0.189654 | 0.0588165 | 0.0140169
#> iteration: 5 | convergence: 0.187515 | 0.0622647 | 0.0163196
#> [1] "Compute neighborhood likelihood..."
#> [1] "Done!"

metrics %>% 
  ggplot(aes(x = lambda, y = cv_like)) +
  geom_point() +
  geom_line() +
  facet_wrap(~k) +
  ylab("CV Likelihood")

metrics %>% 
  group_by(k) %>%
  slice_max(order_by = cv_like, n = 1) %>%
  ungroup() %>%
  mutate(Selected = ifelse(penal_lik == max(penal_lik), "Yes", "No")) %>%
  ggplot(aes(x = k, y = penal_lik, label = lambda)) +
  geom_point() +
  geom_line() +
  geom_label(aes(color = Selected)) +
  ylab("Penalized Likelihood")

We can see that using the recommend procedure, the selected combination is (\(\lambda=15\), \(k=20\)). In practice, we recommend running for a much larger number of iterations (e.g. 1,000) and looking over a larger parameter space.