Persistence Image Vectorization of Persistent Homology — step_vpd_persistence

The function step_vpd_persistence_image() creates a specification of a recipe step that will convert a list-column of 3-column matrices of persistence data to a list-column of 1-row matrices of vectorizations.

step_vpd_persistence_image(
  recipe,
  ...,
  role = "predictor",
  trained = FALSE,
  hom_degree = 0L,
  xseq = NULL,
  xmin = NULL,
  xmax = NULL,
  xlen = NULL,
  xby = NULL,
  yseq = NULL,
  ymin = NULL,
  ymax = NULL,
  ylen = NULL,
  yby = NULL,
  img_sigma = 1,
  columns = NULL,
  keep_original_cols = TRUE,
  skip = FALSE,
  id = rand_id("vpd_persistence_image")
)

Arguments

recipe: A recipe object. The step will be added to the sequence of operations for this recipe.
...: One or more selector functions to choose variables for this step. See selections() for more details.
role: For model terms created by this step, what analysis role should they be assigned? By default, the new columns created by this step from the original variables will be used as predictors in a model.
trained: A logical to indicate if the quantities for preprocessing have been estimated.
hom_degree: The homological degree of the features to be transformed.
xseq: A discretization grid, as an increasing numeric vector. xseq overrides the other x* parameters with a warning.
xmin, xmax, xlen, xby: Limits and resolution of a discretization grid; specify only one of xlen and xby.
yseq: Combined with xseq to form a 2-dimensional discretization grid.
ymin, ymax, ylen, yby: Limits and resolution of a discretization grid; specify only one of ylen and yby.
img_sigma: The standard deviation of the gaussian distribution convolved with persistence diagrams to obtain persistence images.
columns: A character string of the selected variable names. This field is a placeholder and will be populated once prep() is used.
keep_original_cols: A logical to keep the original variables in the output. Defaults to FALSE.
skip: A logical. Should the step be skipped when the recipe is baked by bake()? While all operations are baked when prep() is run, some operations may not be able to be conducted on new data (e.g. processing the outcome variable(s)). Care should be taken when using skip = TRUE as it may affect the computations for subsequent operations.
id: A character string that is unique to this step to identify it.

Value

An updated version of recipe with the new step added to the sequence of any existing operations.

Details

Persistent homology is usually encoded as birth–death pairs (barcodes or diagrams), but the space of persistence data sets does not satisfy convenient statistical properties. Such applications as hypothesis testing and machine learning benefit from transformations of persistence data, often to Hilbert spaces (vector spaces with inner products and induced metrics).

Engine

The persistence image vectorization deploys TDAvec::computePersistenceImage(). See there for definitions and references.

Tuning Parameters

This step has 2 tuning parameters:

hom_degree: Homological degree (type: integer, default: 0L)
img_sigma: Convolved Gaussian standard deviation (type: double, default: 1)

Examples

library(recipes)

# inspect vectorized features
volc_dat <- data.frame(image = I(list(volcano / 10)))
recipe(~ image, data = volc_dat) %>% 
  step_pd_raster(image, method = "link_join") %>% 
  step_vpd_persistence_image(image_pd, hom_degree = 1, img_sigma = 1) %>% 
  print() -> volc_rec
#> 
#> ── Recipe ──────────────────────────────────────────────────────────────────────
#> 
#> ── Inputs 
#> Number of variables by role
#> predictor: 1
#> 
#> ── Operations 
#> • persistent features from a cubical filtration of: image
#> • persistence image of: image_pd
print(volc_rec)
#> 
#> ── Recipe ──────────────────────────────────────────────────────────────────────
#> 
#> ── Inputs 
#> Number of variables by role
#> predictor: 1
#> 
#> ── Operations 
#> • persistent features from a cubical filtration of: image
#> • persistence image of: image_pd
volc_rec %>% 
  prep(training = volc_dat) %>% 
  bake(new_data = volc_dat)
#> # A tibble: 1 × 9,803
#>   image    image_pd image_pd_pi_1 image_pd_pi_2 image_pd_pi_3 image_pd_pi_4
#>   <list>   <list>           <dbl>         <dbl>         <dbl>         <dbl>
#> 1 <dbl[…]> <PHom>        2.58e-52      2.03e-51      1.54e-50      1.12e-49
#> # ℹ 9,797 more variables: image_pd_pi_5 <dbl>, image_pd_pi_6 <dbl>,
#> #   image_pd_pi_7 <dbl>, image_pd_pi_8 <dbl>, image_pd_pi_9 <dbl>,
#> #   image_pd_pi_10 <dbl>, image_pd_pi_11 <dbl>, image_pd_pi_12 <dbl>,
#> #   image_pd_pi_13 <dbl>, image_pd_pi_14 <dbl>, image_pd_pi_15 <dbl>,
#> #   image_pd_pi_16 <dbl>, image_pd_pi_17 <dbl>, image_pd_pi_18 <dbl>,
#> #   image_pd_pi_19 <dbl>, image_pd_pi_20 <dbl>, image_pd_pi_21 <dbl>,
#> #   image_pd_pi_22 <dbl>, image_pd_pi_23 <dbl>, image_pd_pi_24 <dbl>, …

# dimension-reduce using vectorized features
data(permeability_qsar, package = "modeldata")
permeability_qsar %>% 
  transform(perm_cut = cut(permeability, breaks = seq(0, 60, 10))) %>% 
  subset(select = -permeability) %>% 
  tidyr::nest(chem_fp = -perm_cut) %>% 
  print() -> perm_dat
#> # A tibble: 6 × 2
#>   perm_cut chem_fp               
#>   <fct>    <list>                
#> 1 (10,20]  <tibble [20 × 1,107]> 
#> 2 (0,10]   <tibble [110 × 1,107]>
#> 3 (20,30]  <tibble [7 × 1,107]>  
#> 4 (30,40]  <tibble [8 × 1,107]>  
#> 5 (40,50]  <tibble [16 × 1,107]> 
#> 6 (50,60]  <tibble [4 × 1,107]>  
recipe(perm_cut ~ chem_fp, data = perm_dat) %>% 
  step_pd_point_cloud(chem_fp, max_hom_degree = 2) %>% 
  step_vpd_persistence_image(chem_fp_pd, hom_degree = 1, img_sigma = 1) %>% 
  step_pca(starts_with("chem_fp_pd_"), num_comp = 2) %>%
  print() -> perm_rec
#> 
#> ── Recipe ──────────────────────────────────────────────────────────────────────
#> 
#> ── Inputs 
#> Number of variables by role
#> outcome:   1
#> predictor: 1
#> 
#> ── Operations 
#> • persistent features from a Rips filtration of: chem_fp
#> • persistence image of: chem_fp_pd
#> • PCA extraction with: starts_with("chem_fp_pd_")
perm_est <- prep(perm_rec, training = perm_dat)
perm_res <- bake(perm_est, new_data = perm_dat)
# inspect results
tidy(perm_rec)
#> # A tibble: 3 × 6
#>   number operation type                  trained skip  id                       
#>    <int> <chr>     <chr>                 <lgl>   <lgl> <chr>                    
#> 1      1 step      pd_point_cloud        FALSE   FALSE pd_point_cloud_QKjPT     
#> 2      2 step      vpd_persistence_image FALSE   FALSE vpd_persistence_image_Pt…
#> 3      3 step      pca                   FALSE   FALSE pca_4ozuV                
tidy(perm_rec, number = 2)
#> # A tibble: 1 × 3
#>   terms      value id                         
#>   <chr>      <dbl> <chr>                      
#> 1 chem_fp_pd    NA vpd_persistence_image_Pt6Ax
tidy(perm_est, number = 2)
#> # A tibble: 1 × 3
#>   terms      value id                         
#>   <chr>      <dbl> <chr>                      
#> 1 chem_fp_pd    NA vpd_persistence_image_Pt6Ax
# visualize results
with(perm_res, {
  plot(PC1, PC2, type = "n", asp = 1)
  text(PC1, PC2, labels = perm_cut)
})