Complex Polynomial Vectorization of Persistent Homology
Source:R/zzz-step-vpd-complex-polynomial.R
step_vpd_complex_polynomial.RdThe 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.
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 whenprep()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 usingskip = TRUEas 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_pd, 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_pd
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_pd
volc_rec %>%
prep(training = volc_dat) %>%
bake(new_data = volc_dat)
#> # A tibble: 1 × 4
#> image image_pd image_pd_cp_1_1 image_pd_cp_2_1
#> <list> <list> <dbl> <dbl>
#> 1 <dbl [87 × 61]> <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_pd, hom_degree = 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
#> • complex polynomial 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_sJg36
#> 2 2 step vpd_complex_polynomial FALSE FALSE vpd_complex_polynomial_…
#> 3 3 step pca FALSE FALSE pca_z7gvm
tidy(perm_rec, number = 2)
#> # A tibble: 1 × 3
#> terms value id
#> <chr> <dbl> <chr>
#> 1 chem_fp_pd NA vpd_complex_polynomial_RN5gD
tidy(perm_est, number = 2)
#> # A tibble: 1 × 3
#> terms value id
#> <chr> <dbl> <chr>
#> 1 chem_fp_pd NA vpd_complex_polynomial_RN5gD
# visualize results
with(perm_res, {
plot(PC1, PC2, type = "n", asp = 1)
text(PC1, PC2, labels = perm_cut)
})
