Complex Polynomial Vectorization of Persistent Homology — step_vpd_complex

The function step_vpd_complex_polynomial() 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.

Usage

step_vpd_complex_polynomial(
  recipe,
  ...,
  role = "predictor",
  trained = FALSE,
  hom_degree = 0L,
  num_coef = 1L,
  poly_type = "R",
  columns = NULL,
  keep_original_cols = TRUE,
  skip = FALSE,
  id = rand_id("vpd_complex_polynomial")
)

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.
num_coef: The number of coefficients of a convex polynomial fitted to finite persistence pairs.
poly_type: The type of complex polynomial to fit ('R', 'S', or 'T').
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 complex polynomial vectorization deploys TDAvec::computeComplexPolynomial(). See there for definitions and references.

Tuning Parameters

This step has 3 tuning parameters:

hom_degree: Homological degree (type: integer, default: 0L)
num_coef: # Polynomial coefficients (type: integer, default: 1L)
poly_type: Type of polynomial (type: character, default: "R")

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_complex_polynomial(image, hom_degree = 1) %>% 
  print() -> volc_rec
#> 
#> ── Recipe ──────────────────────────────────────────────────────────────────────
#> 
#> ── Inputs 
#> Number of variables by role
#> predictor: 1
#> 
#> ── Operations 
#> • persistent features from a cubical filtration of: image
#> • complex polynomial of: image
print(volc_rec)
#> 
#> ── Recipe ──────────────────────────────────────────────────────────────────────
#> 
#> ── Inputs 
#> Number of variables by role
#> predictor: 1
#> 
#> ── Operations 
#> • persistent features from a cubical filtration of: image
#> • complex polynomial of: image
volc_rec %>% 
  prep(training = volc_dat) %>% 
  bake(new_data = volc_dat)
#> # A tibble: 1 × 3
#>   image           image_cp_1_1 image_cp_2_1
#>   <list>                 <dbl>        <dbl>
#> 1 <PHom [18 × 3]>        -74.2        -82.3

# 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_complex_polynomial(chem_fp, hom_degree = 1) %>% 
  step_pca(starts_with("chem_fp_"), 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
#> • complex polynomial of: chem_fp
#> • PCA extraction with: starts_with("chem_fp_")
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_9OUfU    
#> 2      2 step      vpd_complex_polynomial FALSE   FALSE vpd_complex_polynomial_…
#> 3      3 step      pca                    FALSE   FALSE pca_He0x9               
tidy(perm_rec, number = 2)
#> # A tibble: 1 × 3
#>   terms   value id                          
#>   <chr>   <dbl> <chr>                       
#> 1 chem_fp    NA vpd_complex_polynomial_RR0yg
tidy(perm_est, number = 2)
#> # A tibble: 1 × 3
#>   terms   value id                          
#>   <chr>   <dbl> <chr>                       
#> 1 chem_fp    NA vpd_complex_polynomial_RR0yg
# visualize results
with(perm_res, {
  plot(PC1, PC2, type = "n", asp = 1)
  text(PC1, PC2, labels = perm_cut)
})