Persistent homology of point clouds

The function step_pd_point_cloud() creates a specification of a recipe step that will convert compatible data formats (distance matrices, coordinate matrices, or time series) to 3-column matrix representations of persistence diagram data. The input and output must be list-columns.

step_pd_point_cloud(
  recipe,
  ...,
  role = "persistence diagram",
  trained = FALSE,
  filtration = "Rips",
  max_hom_degree = 1L,
  radius_max = NULL,
  diameter_max = NULL,
  field_order = 2L,
  engine = NULL,
  columns = NULL,
  keep_original_cols = TRUE,
  skip = FALSE,
  id = rand_id("pd_point_cloud")
)

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.
filtration: The type of filtration from which to compute persistent homology; one of "Rips", "Vietoris" (equivalent), or "alpha".
max_hom_degree, radius_max, diameter_max, field_order: Parameters passed to persistence engines.
engine: The computational engine to use (see 'Details'). Reasonable defaults are chosen based on filtration.
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 (PH) is a tool of algebraic topology to extract features from data whose persistence measures their robustness to scale. The computation relies on a sequence of maps between discrete topological spaces (usually a filtration comprising only inclusions) constructed from the data.

PH of Point Clouds

The PH of a point cloud arises from a simplicial filtration (usually Vietoris–Rips, Čech, or alpha) along an increasing distance threshold.

Ripser is a highly efficient implementation of PH on a point cloud (a finite metric space) via the Vietoris–Rips filtration and is ported to R through ripserr. TDA calls the Dionysus, PHAT, and GUDHI libraries to compute PH via Vietoris–Rips and alpha filtrations. The filtration parameter controls the choice of filtration while the engine parameter allows the user to manually select an implementation.

Both engines accept data sets in distance matrix, coordinate matrix, and data frame formats. While ripserr computes PH for time series data, this is not currently supported in tdarec.

The max_hom_degree argument determines the highest-dimensional features to be calculated. Either diameter_max (preferred) or radius_max can be used to bound the distance threshold along which PH is computed. The field_order argument should be prime and will be the order of the field of coefficients used in the computation. In most applications, only max_hom_degree will be tuned, and to at most 3L.

Tuning Parameters

This step has 1 tuning parameter(s):

max_hom_degree: Maximum Homological Degree (type: integer, default: 1)

Examples

roads <- data.frame(dist = I(list(eurodist, UScitiesD * 1.6)))

ph_rec <- recipe(~ ., data = roads) %>% 
  step_pd_point_cloud(dist, max_hom_degree = 1, filtration = "Rips")
ph_prep <- prep(ph_rec, training = roads)
ph_res <- bake(ph_prep, roads)

tidy(ph_rec, number = 1)
#> # A tibble: 1 × 3
#>   terms value id                  
#>   <chr> <dbl> <chr>               
#> 1 dist     NA pd_point_cloud_xwT1j
tidy(ph_prep, number = 1)
#> # A tibble: 1 × 3
#>   terms value id                  
#>   <chr> <dbl> <chr>               
#> 1 dist     NA pd_point_cloud_xwT1j

ops <- par(mfrow = c(1, 2), mar = c(2, 2, 0, 0) + 0.1)
for (i in seq(nrow(ph_res))) {
  with(ph_res$dist_pd[[i]], plot(
    x = birth, y = death, pch = dimension + 1, col = dimension + 1,
    xlab = NA, ylab = "", asp = 1
  ))
}

par(ops)

with_max <- recipe(~ ., data = roads) %>% 
  step_pd_point_cloud(dist, max_hom_degree = 1, diameter_max = 200)
with_max <- prep(with_max, training = roads)
bake(with_max, roads)
#> # A tibble: 2 × 2
#>   dist         dist_pd       
#>   <list>       <list>        
#> 1 <dist [210]> <PHom [2 × 3]>
#> 2 <dist [45]>  <PHom [0 × 3]>